Hecke algebras for inner forms of p-adic special linear groups

HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS arXiv:1406.6861v1 [math.RT] 26 Jun 2014 ANNE-MARIE AUBERT, PAUL BAUM, ROGER PLYMEN, AND MAARTEN SOLLEVELD Abstract. Let F be a non-archimedean local field and let G♯ be the group of F rational points of an inner form of SLn . We study Hecke algebras for all Bernstein components of G♯ , via restriction from an inner form G of GLn (F ). For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth G♯ -representations. This algebra comes from an idempotent in the full Hecke algebra of G♯ , and the idempotent is derived from a type for G. We show that the Hecke algebras for Bernstein components of G♯ are similar to affine Hecke algebras of type A, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group. Contents Introduction 1. Notations and conventions 2. Restricting representations 2.1. Restriction to the derived group 2.2. Restriction of Bernstein components 2.3. The intermediate group 3. Morita equivalences 3.1. Construction of a particular idempotent 3.2. Descent to a Levi subgroup 3.3. Passage to the derived group 4. The structure of the Hecke algebras 4.1. Hecke algebras for general linear groups 4.2. Projective normalizers 4.3. Hecke algebras for the intermediate group 4.4. Hecke algebras for the derived group 5. Examples References Date: June 28, 2014. 2010 Mathematics Subject Classification. 20G25, 22E50. Key words and phrases. representation theory, division algebra, Hecke algebra, types. The second author was partially supported by NSF grant DMS-1200475. 1 2 6 9 9 11 14 17 18 24 28 36 36 39 46 51 54 59 2 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Introduction Let F be a non-archimedean local field and let D be a division algebra, of dimension d2 over its centre F . Then G = GLm (D) is the group of F -rational points of an inner form of GLn , where n = md. We will say simply that G is an inner form of GLn (F ). Its derived group G♯ , the kernel of the reduced norm map G → F × , is an inner form of SLn (F ). Every inner form of SLn (F ) looks like this. Since the appearance of the important paper [HiSa] there has been a surge of interest in these groups, cf. [ChLi, ChGo, ABPS2]. In this paper we continue our investigations of the (complex) representation theory of inner forms of SLn (F ). Following the Bushnell–Kutzko approach [BuKu3], we study algebras associated to idempotents in the Hecke algebra of G♯ . The main idea of this approach is to understand Bernstein components for a reductive p-adic group better by constructing types and making the ensuing Hecke algebras explicit. It turns out that for the groups under consideration, while it is really hard to find types, the appropriate Hecke algebras are accessible via different techniques. Our starting point is the construction of types for all Bernstein components of G by Sécherre–Stevens [SéSt1, SéSt2]. We consider the Hecke algebra of such a type, which is described in [Séc]. In several steps we modify this algebra to one whose module category is equivalent to a union of some Bernstein blocks for G♯ . Let discuss our strategy and the main result in more detail. We fix a parabolic subgroup P ⊂ G with Levi factor L. Given an inertial equivalence class s = [L, ω]G , let Reps (G) denote the corresponding Bernstein block of the category of smooth complex G-representations, and let Irrs (G) denote the set of irreducible objects in Reps (G). Let Irrs (G♯ ) be the set of irreducible G♯ -representations s s ♯ that are subquotients of ResG G♯ (π) for some π ∈ Irr (G). We define Rep (G ) as the collection of G♯ -representations all whose irreducible subquotients lie in Irrs (G♯ ). We want to investigate the category Reps (G♯ ). It is a product of finitely many Bernstein blocks for G♯ : Y ♯ (1) Reps (G♯ ) = Rept (G♯ ). ♯ t ≺s ♯ We note that the Bernstein components Irrt (G♯ ) which are subordinate to one s form precisely one class of L-indistinguishable components: every L-packet for G♯ which intersects one of them intersects them all. The structure of Reps (G) is largely determined by the torus Ts and the finite group Ws associated by Bernstein to s. Recall that the Bernstein torus of s is Ts = {ω ⊗ χ | χ ∈ Xnr (L)} ⊂ Irr(L), where Xnr (L) denotes the group of unramified characters of L. The finite group Ws equals NM (L)/L for a suitable Levi subgroup M ⊂ G containing L. For this particular reductive p-adic group Ws is always a Weyl group (in fact a direct product of symmetric groups), but for G♯ more general finite groups are needed. We also have to take into account that we are dealing with several Bernstein components simultaneously. Let H(G) be the Hecke algebra of G and H(G)s its two-sided ideal corresponding to the Bernstein block Reps (G). Similarly, let H(G♯ )s be the two-sided ideal of HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS H(G♯ ) corresponding to Reps (G♯ ). Then Y H(G♯ )s = ♯ 3 ♯ t ≺s H(G♯ )t . ♯ Of course we would like to determine H(G♯ )t , but it turns out that H(G♯ )s is much easier to study. So our main goal is an explicit description of H(G♯ )s up to Morita ♯ equivalence. From that the subalgebras H(G♯ )t can in principle be extracted, as ♯ maximal indecomposable subalgebras. We note that sometimes H(G♯ )s = H(G♯ )t , see Examples 5.2, 5.4. From [SéSt1] we know that there exists a simple type (K, λ) for [L, ω]M . By [SéSt2] it has a G-cover (KG , λG ). We denote the associated central idempotent of H(K) by eλ , and similarly for other irreducible representations. There is an affine Hecke algebra H(Ts , Ws , qs ), a tensor product of affine Hecke algebras of type GLe , such that (2) eλG H(G)eλG ∼ = eλ H(M )eλ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vλ ), and these algebras are Morita equivalent with H(G)s . An important role in the restrictions of representations from H(G)s to H(G♯ )s is played by the group X G (s) := {γ ∈ Irr(G/G♯ Z(G)) | γ ⊗ IPG (ω) ∈ Reps (G)}. It acts on H(G) by pointwise multiplication of functions G → C. For the restriction process we need an idempotent that is invariant under X G (s). To that end we replace λG by the sum of the representations γ ⊗ λG with γ ∈ X G (s), which we call µG . Then (2) remains valid with µ instead of λ, but of course Vµ is reducible as a representation of K. Let eµG♯ ∈ H(G♯ ) be the restriction of eµG : G → C to G♯ . Up to a scalar factor it is also the restriction of eλG to G♯ . We normalize the Haar measures in such a way that eµG♯ is idempotent. For any G♯ -representation V, eµG♯ V is the space of vectors in V on which KG ∩G♯ acts as some multiple of the (reducible) representation λG |KG ∩G♯ . Then eµG♯ H(G♯ )eµG♯ is a nice subalgebra of H(G♯ )s , but in general it is not Morita equivalent with H(G♯ )s . There is only an equivalence between the module category Q ♯ of eµG♯ H(G♯ )eµG♯ and t♯ Rept (G♯ ), where t♯ runs over some, but not necessarily all, inertial equivalence classes ≺ s. To see the entire category Reps (G♯ ) we need finitely many isomorphic but mutually orthogonal algebras aeµG♯ a−1 H(G♯ )aeµG♯ a−1 with a ∈ G. To formulate our main result precisely, we need also the groups X L (s) = {γ ∈ Irr(L/L♯ Z(G)) | γ ⊗ ω ∈ [L, ω]L }, Ws♯ = {w ∈ NG (L) | ∃γ ∈ Irr(L/L♯ Z(G)) : w(γ ⊗ ω) ∈ [L, ω]L }, R♯s = Ws♯ ∩ NG (P ∩ M )/L, X L (ω, Vµ ) = {γ ∈ Irr(L/L♯ ) | there exists an L-isomorphism ω → ω ⊗ γ −1 which induces the identity on Vµ }. 4 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Here L♯ = L ∩ G♯ , so L/L♯ ∼ = G/G♯ ∼ = F ×. We observe that R♯s is naturally isomorphic to X G (s)/X L (s), and that Ws♯ = Ws ⋊R♯s (see Lemmas 2.4 and 2.3). One can regard Ws♯ as the Bernstein group for Reps (G♯ ). See also examples 4 and 5.2. Theorem 1. The algebra H(G♯ )s is Morita equivalent with a direct sum of |X L (ω, Vµ )| copies of eµG♯ H(G♯ )eµG♯ . The latter algebra is isomorphic with
X L (s)/X L (ω,Vµ )Xnr (L/L♯ Z(G)) ⋊ R♯s , H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) where Ts♯ = Ts /Xnr (L/L♯ ). The actions of the groups X L (s) and R♯s come from automorphisms of Ts ⋊ Ws and projective transformations of Vµ . The projective actions of X L (s) and R♯s on Vµ are always linear in the split case G = GLn (F ), but not in general, see examples 5.3 and 5.4. Contrary to what one might expect from Theorem 1, the Bernstein torus for ♯ Rept (G♯ ) is not always Ts♯ , see example 5.1. In general one has to divide by a finite subgroup of Ts♯ coming from X L (s). Of course the above has already been done for SLn (F ) itself, see [BuKu1, BuKu2, GoRo1, GoRo2]. Indeed, for SLn (F ) our work has a large intersection with these papers. But the split case is substantially easier than the non-split case, for example because every irreducible representation of GLn (F ) restricts to a representation of SLn (F ) without multiplicities. Therefore our methods are necessarily different from those of Bushnell–Kutzko and Goldberg–Roche, even if our proofs are considered only for SLn (F ). It is interesting to compare Theorem 1 for SLn (F ) with the main results of [GoRo2]. Our description of the Hecke algebras is more explicit, thanks to considering the entire packet Reps (G♯ ) of Bernstein blocks simultaneously. In [GoRo2, §11] some 2-cocycle pops up in the Hecke algebras, which Goldberg–Roche expect to be trivial. From Theorem 1 one can deduce that it is indeed trivial. Let us describe the contents of the paper in more detail. We start Section 2 with recalling a few results about restriction of representations from G to G♯ . Then we discuss what happens when one restricts an entire Bernstein component of representations at once. This leads to several finite groups that play a role in this process. It turns out to be advantageous to restrict from G to G♯ in two steps, via G♯ Z(G). This intermediate group is of finite index in G if the characteristic of F does not divide n. Otherwise [G : G♯ Z(G)] = ∞ but, when studying only Reps (G), one can apply the same techniques as for a group extension of finite index. Restriction from G♯ Z(G) to G♯ is straightforward, so everything comes down to understanding the decomposition of representations and Bernstein components of G upon restriction to G♯ Z(G). For any subgroup H ⊂ G we write H ♯ = H ∩ G♯ . The correct analogue of Ws for Reps (G) combines Weyl groups and characters of the Levi subgroup L that are trivial on L♯ Z(G): Stab(s) := {(w, γ) ∈ W (G, L) × Irr(L/L♯ Z(G)) | w(γ ⊗ ω) ∈ [L, ω]L }. HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 5 This group acts naturally on L-representations by ((w, γ)π)(l) = w(γ ⊗ π)(l) := γ(l)π(w−1 lw). From another angle Stab(s) can be considered as the generalization, for the inertial class s, of some groups associated to a single G-representation in [ChLi, ChGo]. Its relevance is confirmed by the following result. Theorem 2. [see Theorem 2.7.] Let χ1 , χ2 ∈ Xnr (L). The following are equivalent: G G G (i) ResG G♯ Z(G) (IP (ω⊗χ1 )) and ResG♯ Z(G) (IP (ω⊗χ2 )) have a common irreducible subquotient; G G G (ii) ResG G♯ Z(G) (IP (ω ⊗ χ1 )) and ResG♯ Z(G) (IP (ω ⊗ χ2 )) have the same irreducible constituents, counted with multiplicity; (iii) ω ⊗ χ1 and ω ⊗ χ2 belong to the same Stab(s)-orbit. In Section 3 we build the idempotents that allow us to replace H(G♯ )s by simpler, Morita equivalent algebras. We denote Morita equivalence by ∼M . The main idea is that G H(G♯ )s ∼M (H(G)s )X (s)Xnr (G) . We cannot apply this directly to eλG H(G)eλG , but we exhibit a larger idempotent e♯λG which overcomes this problem. Theorem 3. [see Theorems 3.15 and 3.16.] There are Morita equivalences e♯λG H(G)e♯λG ∼M eλG H(G)eλG ∼M H(G)s , (e♯λG H(G)e♯λG )X G (s) (e♯λG H(G)e♯λG )X G (s)X Here e♯λ e♯λ G♯ Z(G) G♯ ∼M e♯λ G♯ Z(G) nr (G) H(G♯ Z(G))e♯λ G♯ Z(G) ∼M e♯λ ♯ H(G♯ )e♯λ G G♯ ∼M H(G♯ Z(G))s ∼M H(G♯ )s . ∈ H(G♯ ) is the restriction of e♯λG : G → C to G♯ , and similarly for . In another direction we want to simplify things by working in the Hecke algebra of M , the smallest Levi subgroup of G such that Ws = NM (L)/L. A technically complicated step is the construction of an idempotent esM ∈ H(M ) such that (3) e♯λ G♯ Z(G) H(G♯ Z(G))e♯λ G♯ Z(G) ∼ = (esM H(M )esM )XL (s) ⋊ R♯s . Section 4 focuses on the structure of the algebras (3). Using the work of Sécherre [Séc] we first show that esM H(M )esM ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ) ⊗ M|X L (ω,Vµ )| (C). Then we analyse the actions of X L (s) and R♯s on the latter algebra. This to leads to a version of Theorem 2 for H(G♯ Z(G))s (Theorem 4.12), from which we deduce our main result. On the way we also take some steps towards types for G♯ , but we encounter serious obstructions. Finally, in Section 5 we provide some examples of the phenomena that can occur. Let us already give one of them: 6 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Example 4. Let ζ be a ramified character of D × of order 3 and take G = GL6 (D), L = GL1 (D)6 , ω = 1 ⊗ 1 ⊗ ζ ⊗ ζ ⊗ ζ 2 ⊗ ζ 2 . For L = GL2 (D)3 , s = [L, ω]G we have Ts = Xnr (L) ∼ = (C× )6 , Ws = W (M, L) ∼ = (S2 )3 . Furthermore X L (ω) = {1} and X G (IPG (ω)) = {1, ζ, ζ 2 }, R♯s = h(135)(246)i ⊂ S6 ∼ = W (G, L). See Subsections 2.1 and 2.2 for the definitions of these groups. The stabilizer of ω in Stab(s) is generated by ((135)(246), ζ) and Stab(s) = Stab(ω)Ws Xnr (G/Z(G)). Let H(GL2 , q) denote an affine Hecke algebra of type GL2 . By Theorems 4.12 and 4.14 there are Morita equivalences H(G♯ Z(G))s ∼M (H(GL2 , q)⊗3 )Xnr (G/Z(G)) ⋊ R♯s , H(G♯ )s ∼M (H(GL2 , q)⊗3 )Xnr (G) ⋊ R♯s . Now we see that s gives rise to a unique inertial class t♯ for G♯ . Hence Wt♯ = Ws♯ = Ws ⋊ R♯s ) Ws . That is, the finite group associated by Bernstein to t♯ is strictly larger than the finite group for s. Interestingly, some of the algebras that turn up do not look like affine Hecke algebras. In the literature there was hitherto (to the best of our knowledge) only one example of a Hecke algebra of a type which was not Morita equivalent to the a crossed product of an affine Hecke algebra with a finite group, namely [GoRo2, §11.8]. But in several cases of Theorem 1 the part EndC (Vµ ) plays an essential role, and it cannot be removed via some equivalence. Hence these algebras are further away from affine Hecke algebras than any previously known Hecke algebras related to types. See especially example 5.4. The last three examples indicate why it is hard to construct types for G♯ from types for G. Acknowledgements. emails and discussions. The authors thank Shaun Stevens for several helpful 1. Notations and conventions We start with some generalities, to fix the notations. Good sources for the material in this section are [Ren, BuKu3]. Let G be a connected reductive group over a local non-archimedean field. All our representations are tacitly assumed to be smooth and over the complex numbers. We write Rep(G) for the category of such G-representations and Irr(G) for the collection of isomorphism classes of irreducible representations therein. Let P be a parabolic subgroup of G with Levi factor L. The “Weyl” group of L is W (G, L) = NG (L)/L. It acts on equivalence classes of L-representations π by (w · π)(g) = π(w̄gw̄−1 ), HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 7 where w̄ ∈ NG (L) is a chosen representative for w ∈ W (G, L). We write Wπ = {w ∈ W (G, L) | w · π ∼ = π}. Let ω be an irreducible supercuspidal L-representation. The inertial equivalence class s = [L, ω]G gives rise to a category of smooth G-representations Reps (G) and a subset Irrs (G) ⊂ Irr(G). Write Xnr (L) for the group of unramified characters L → C× . Then Irrs (G) consists of all irreducible irreducible constituents of the parabolically induced representations IPG (ω ⊗ χ) with χ ∈ Xnr (L). We note that IPG always means normalized, smooth parabolic induction from L via P to G. The set IrrsL (L) with sL = [L, ω]L can be described explicitly, namely by (4) Xnr (L, ω) = {χ ∈ Xnr (L) : ω ⊗ χ ∼ = ω}, (5) IrrsL (L) = {ω ⊗ χ : χ ∈ Xnr (L)/Xnr (L, ω)}. Several objects are attached to the Bernstein component Irrs (G) of Irr(G) [BeDe]. Firstly, there is the torus Ts := Xnr (L)/Xnr (L, ω), which is homeomorphic to IrrsL (L). Secondly, we have the groups NG (sL ) ={g ∈ NG (L) | g · ω ∈ IrrsL (L)} ={g ∈ NG (L) | g · [L, ω]L = [L, ω]L }, Ws :={w ∈ W (G, L) | w · ω ∈ IrrsL (L)} = NG (sL )/L. Of course Ts and Ws are only determined up to isomorphism by s, actually they depend on sL . To cope with this, we tacitly assume that sL is known when talking about s. The choice of ω ∈ IrrsL (L) fixes a bijection Ts → IrrsL (L), and via this bijection the action of Ws on IrrsL (L) is transferred to Ts . The finite group Ws can be thought of as the ”Weyl group” of s, although in general it is not generated by reflections. Let Cc∞ (G) be the vector space of compactly supported locally constant functions G → C. The choice of a Haar measure on G determines a convolution product * on Cc∞ (G). The algebra (Cc∞ (G), ∗) is known as the Hecke algebra H(G). There is an equivalence between Rep(G) and the category Mod(H(G)) of H(G)-modules V such that H(G) · V = V . We denote the collection of inertial equivalence classes for G by B(G). The Bernstein decomposition Y Rep(G) = Reps (G) s∈B(G) induces a factorization in two-sided ideals Y H(G) = s∈B(G) H(G)s . Let K be a compact open subgroup of K and let (λ, Vλ ) be an irreducible Krepresentation. Let eλ ∈ H(K) be the associated central idempotent and write Repλ (G) = {(π, V ) ∈ Rep(G) | H(G)eλ · V = V }. Clearly eλ H(G)eλ is a subalgebra of H(G), and V 7→ eλ · V defines a functor from Rep(G) to Mod(eλ H(G)eλ ). By [BuKu3, Proposition 3.3] this functor restricts to an equivalence of categories Repλ (G) → Mod(eλ H(G)eλ ) if and only if Repλ (G) is 8 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD closed under taking G-subquotients. Moreover, in that case there are finitely many inertial equivalence classes s1 , . . . sκ such that Repλ (G) = Reps1 (G) × · · · × Repsκ (G). One calls (K, λ) a type for {s1 , . . . , sκ }, or an s1 -type if κ = 1. To a type (K, λ) one associates the algebra H(G, λ) := {f : G → EndC (Vλ∨ ) | supp(f ) is compact, f (k1 gk2 ) = λ∨ (k1 )f (g)λ∨ (k2 ) ∀g ∈ G, k1 , k2 ∈ K}. Here (λ∨ , Vλ∨ ) is the contragredient of (λ, Vλ ) and the product is convolution of functions. By [BuKu3, (2.12)] there is a canonical isomorphism (6) eλ H(G)eλ ∼ = H(G, λ) ⊗C EndC (Vλ ). From now on we discuss things that are specific for G = GLm (D), where D is a central simple F -algebra. We write dimF (D) = d2 . Every P Levi subgroup L of G Q is isomorphic to j GLm̃j (D) for some m̃j ∈ N with j m̃j = m. Hence every irreducible L-representation ω can be written as ⊗j ω̃j with ω̃j ∈ Irr(GLm̃j (D)). Then ω is supercuspidal if and only if every ω̃j is so. As above, we assume that this is the case. Replacing (L, ω) by an inertially equivalent pair allows us to make the following simplifying assumptions: Condition 1.1. • if m̃i = m̃j and [GLm̃j (D), ω̃i ]GLm̃j (D) = [GLm̃j (D), ω̃j ]GLm̃j (D) , then ω̃i = ω̃j ; Q • ω = Qi ωi⊗ei , such Q that ωi and ωj are not inertially equivalent if i 6= j; • L = i Lei i = i GLmi (D)ei , embedded diagonally in GLm (D) such that factors Li with the same (mi , ei ) are in subsequent positions; • as representatives for the elements of W (G, L) we take permutation matrices; • P is the parabolic subgroup of G generated by L and the upper triangular matrices; • if mi = mj , ei = ej and ωi is isomorphic to ωj ⊗ γ for some character γ of GLmi (D), then ωi = ωj ⊗ γχ for some χ ∈ Xnr (GLmi (D)). We remark that these conditions are natural generalizations of [GoRo2, §1.2] to our setting. Most of the time we will not need the conditions for stating the results, but they are useful in many proofs. Under Conditions 1.1 we define Y ej
Y Y Y Lj = GLmi ei (D), Mi = ZG (7) M= i i i j6=i a Levi subgroup of G containing L. For s = [L, ω]G we have Y Y (8) Ws = W (M, L) = NM (L)/L = NMi (Lei i )/Lei i ∼ Sei , = i i a direct product of symmetric groups. Writing si = [Li , ωi ]Li , the torus associated to s becomes Y Ts = (Tsi )ei , (9) i (10) Tsi = Xnr (Li )/Xnr (Li , ωi ). HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 9 By our choice of representatives for W (G, L), ωi⊗ei is stable under NMi (Lei i )/Lei i ∼ = Sei . The action of Ws on Ts is just permuting coordinates in the standard way and (11) Ws = Wω . 2. Restricting representations 2.1. Restriction to the derived group. We will study the restriction of representations of G = GLm (D) to its derived group G♯ = GLm (D)der . For subgroups H ⊂ G we will write H ♯ = H ∩ G♯ . Recall that the reduced norm map Nrd: Mm (D) → F induces a group isomorphism Nrd : G/G♯ → F × . We start with some important relations between representations of G and G♯ , which were proven both by Tadić and by Bushnell–Kutzko. Proposition 2.1. (a) Every irreducible representation of G♯ appears in an irreducible representation of G. (b) For π, π ′ ∈ Irr(G) the following are equivalent: G ′ (i) ResG G♯ (π) and ResG♯ (π ) have a common irreducible subquotient; G ′ ∼ (ii) ResG G♯ (π) = ResG♯ (π ); (iii) there is a γ ∈ Irr(G/G♯ ) such that π ′ ∼ = π ⊗ γ. ♯ (c) The restriction of (π, V ) ∈ Irr(G) to G is a finite direct sum of irreducible G♯ -representations, each one appearing with the same multiplicity. (d) Let (π ′ , V ′ ) be an irreducible G♯ -subrepresentation of (π, V ). Then the stabilizer in G of V ′ is a open, normal, finite index subgroup Hπ ⊂ G which contains G♯ and the centre of G. Proof. All these results can be found in [Tad, §2], where they are in fact shown for any reductive group over a local non-archimedean field. For G = GLn (F ), these statements were proven in [BuKu1, Propositions 1.7 and 1.17] and [BuKu2, Proposition 1.5]. The proofs in [BuKu1, BuKu2] also apply to G = GLm (D).  Let π ∈ Irr(G). By Proposition 2.1.d (12) EndG♯ (V ) = EndHπ (V ), which allows us to use [HiSa, Chapter 2] and [GeKn, Section 2] (which is needed for [HiSa]). We put X G (π) := {γ ∈ Irr(G/G♯ ) | π ⊗ γ ∼ = π}. As worked out in [HiSa, Chapter 2], this group governs the reducibility of ResG G♯ (π). G (We will use this definition of X (π) more generally if π ∈ Rep(G) admits a central character.) By (12) every element of X G (π) is trivial on Hπ , so X G (π) is finite. Via the local Langlands correspondence for G, the group X G (π) corresponds to the geometric R-group of the L-packet for G♯ obtained from ResG G♯ (π), see [ABPS2, §3]. We note that X G (π) ∩ Xnr (G) = Xnr (G, π). For every γ ∈ X G (π) there exists a nonzero intertwining operator (13) I(γ, π) ∈ HomG (π ⊗ γ, π) = HomG (π, π ⊗ γ −1 ), 10 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD which is unique up to a scalar. As G♯ ⊂ ker(γ), I(γ, π) can also be considered as an element of EndG♯ (π). As such, these operators determine a 2-cocycle κπ by I(γ, π) ◦ I(γ ′ , π) = κπ (γ, γ ′ )I(γγ ′ , π). (14) By [HiSa, Lemma 2.4] they span the G♯ -intertwining algebra of π: G ∼ EndG♯ (ResG G♯ π) = C[X (π), κπ ], (15) where the right hand side denotes the twisted group algebra of X G (π). By [HiSa, Corollary 2.10] M ∼ HomC[X G (π),κπ ] (ρ, π) ⊗ ρ (16) ResG G♯ π = ρ∈Irr(C[X G (π),κπ ]) as representations of G♯ × X G (π). Let P be a parabolic subgroup of G with Levi factor L. The inclusion L → G induces isomorphisms L/L♯ → G/G♯ ∼ = F ×. (17) Let ω ∈ Irr(L) be supercuspidal and unitary. Using (17) we can identify the G♯ representations G ResG G♯ (IP (ω)) ♯ and IPG♯ (ResL L♯ (ω)), which yields an inclusion X L (ω) → X G (IPG (ω)). Every intertwining operator I(γ, ω) for γ ∈ X L (ω) induces an intertwining operator I(γ, IPG (ω)) := IPG (I(γ, ω)) ∈ HomG (γ ⊗ IPG (ω), IPG (ω)), (18) for γ as an element of X G (IPG (ω)). We warn that, even though X L (ω) is finite abelian and ω is supercuspidal, it is still possible that the 2-cocycle κω is nontrivial and that L ∼ EndL♯ (ResL L♯ (ω)) = C[X (ω), κω ] is noncommutative, see [ChLi, Example 6.3.3]. We introduce the groups (19) (20) Wω♯ = {w ∈ W (G, L) | ∃γ ∈ Irr(L/L♯ ) | w · (γ ⊗ ω) ∼ = ω}, ♯ Stab(ω) = {(w, γ) ∈ W (G, L) × Irr(L/L ) | w · (γ ⊗ ω) ∼ = ω}. Notice that the actions of W (G, L) and Irr(L/L♯ ) on Irr(L) commute because every element of Irr(L/L♯ ) extends to a character of G which is trivial on the derived subgroup of G. Clearly Wω × X L (ω) is a normal subgroup of Stab(ω) and there is a short exact sequence 1 → X L (ω) → Stab(ω) → Wω♯ → 1. (21) By [ChLi, Proposition 6.2.2] the projection of Stab(ω) on the second coordinate gives rise to a short exact sequence 1 → Wω → Stab(ω) → X G (IPG (ω)) → 1 (22) and the group (23) R♯ω := Stab(ω)/(Wω × X L (ω)) ∼ = X G (IPG (ω))/X L (ω) ∼ = Wω♯ /Wω HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 11 is naturally isomorphic to the “dual R-group” of the L-packet for G♯ obtained from G ResG G♯ (IP (ω)). We remark that, with the method of Lemma 2.3.c, it is also possible to realize R♯ω as a subgroup of Stab(ω). By (16) intertwining operators associated to elements of Stab(ω) span EndG♯ (IPG (ω)) whenever ω is supercuspidal and unitary. By [ABPS1, Theorem 1.6.a] that holds more generally for IPG (ω ⊗χ) when χ is in Langlands position with respect to P . The group Stab(ω) also acts on the set of irreducible L♯ -representations appearing L ♯ in ResL L♯ (ω). For an irreducible subrepresentation σ of ResL♯ (ω) [ChLi, Proposition 6.2.3] says that Wω ⊂ Wσ♯ ⊂ Wω♯ (24) ♯ and that the analytic R-group of IPG♯ (σ ♯ ) is Rσ♯ := Wσ♯ /Wω , the stabilizer of σ ♯ in R♯ω . It is possible that Wσ♯ 6= Wω♯ and Rσ♯ 6= R♯ω , see [ChLi, Example 6.3.4]. In view of [ABPS1, Section 1] the above results remain valid if ω ∈ Irr(L) is assumed to be supercuspidal but not necessarily unitary. Just one modification is required: if IPG (ω) is reducible, one should consider the L-packet for G♯ obtained from the (unique) Langlands constituent of IPG (ω). 2.2. Restriction of Bernstein components. Next we study the restriction of an entire Bernstein component Irrs (G) to G♯ . Let Irrs (G♯ ) be the set of irreducible G♯ -representations that are subquotients of s ResG G♯ (π) for some π ∈ Irr (G). Lemma 2.2. Irrs (G♯ ) is a union of finitely many Bernstein components for G♯ . Proof. Consider any π ♯ ∈ Irrs (G♯ ). It is a subquotient of ♯ L G G ResG G♯ (IP (ω ⊗ χ1 )) = IP ♯ (ResL♯ (ω ⊗ χ1 )) for some χ1 ∈ Xnr (L). Choose an irreducible summand σ1 of the supercuspidal ♯ ♯ G♯ L♯ -representation ResL L♯ (ω ⊗ χ1 ), such that π is a subquotient of IP ♯ (σ1 ). Then π ♯ lies in the Bernstein component Irr[L ,σ1 ]G♯ (G♯ ). Any unramified character of χ♯2 of L♯ lifts to an unramified character of L, say χ2 . Now ♯ ♯ ♯ G G L G IPG♯ (ResL L♯ (σ1 ) ⊗ χ2 ) ⊂ IP ♯ (ResL♯ (ω ⊗ χ1 χ2 )) = ResG♯ (IP (ω ⊗ χ1 χ2 )), ♯ which shows that all irreducible subquotients of IPG♯ (ResL L♯ (σ1 ) ⊗ χ2 ) belong to ♯ ♯ Irrs (G♯ ). It follows that Irr[L ,σ1 ]G♯ (G♯ ) ⊂ Irrs (G♯ ). ♯ The above also shows that any inertial equivalence class t♯ with Irrt (G♯ ) ⊂ Irrs (G♯ ) must be of the form (25) t♯ = [L♯ , σ2 ]G♯ for some irreducible constituent σ2 of ResL L♯ (ω ⊗ χ2 ). So up to an unramified twist σ2 is an irreducible constituent of ResL (ω). Now Proposition 2.1.c shows that there L♯ ♯ are only finitely many possibilities for t .  12 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD ♯ Motivated by this lemma, we write t♯ ≺ s if Irrt (G♯ ) ⊂ Irrs (G♯ ). In other words, [ ♯ (26) Irrs (G♯ ) = Irrt (G♯ ). ♯ t ≺s The last part of the proof of Lemma 2.2 shows that every t♯ ≺ s is of the form [L♯ , σ ♯ ]G♯ for some irreducible constituent σ ♯ of ResL L♯ (ω). Recall from (16) that con♯ stituents σ as above are parametrized by irreducible representations of the twisted group algebra C[X L (ω), κω ]. However, non-isomorphic σ ♯ may give rise to the same inertial equivalence class t♯ for G♯ . It is quite difficult to determine the Bernstein tori Tt♯ precisely. The finite group associated by Bernstein to t♯ = [L♯ , σ ♯ ]G♯ is its stabilizer in W (G, L) = W (G♯ , L♯ ): Wt♯ = {w ∈ W (G, L) | w · σ ♯ ∈ [L♯ , σ ♯ ]L♯ }. As the different σ ♯ are M -conjugate, they all produce the same group Wt♯ . So it depends only on [M, σ]M . It is quite possible that Wt♯ is strictly larger than Ws , we already saw this in Example 4. First steps to study such cases were sketched (for SLn (F )) in [BuKu2, §9]. For every w ∈ Wt♯ , Proposition 2.1.b (for L♯ ) guarantees the existence of a γ ∈ Irr(L/L♯ ) such that w(ω) ⊗ γ ∈ [L, ω]L . Up to an unramified character γ has to be trivial on Z(G), so we may take γ ∈ Irr(L/L♯ Z(G)). Hence Wt♯ is contained in the group Ws♯ := {w ∈ W (G, L) | ∃γ ∈ Irr(L/L♯ Z(G)) such that w(γ ⊗ ω) ∈ [L, ω]L } By (24) the subgroup Wω fixes every [L♯ , σ ♯ ]L♯ , so Wω ⊂ Wt♯ and (27) StabW ♯ /Wω ([L♯ , σ ♯ ]L♯ ) = Wt♯ /Wω ⊃ Wσ♯ /Wω . s Lemma 2.3. (a) Ws is a normal subgroup of Ws♯ which fixes all the [L♯ , σ ♯ ]L♯ with [L♯ , σ ♯ ]G♯ ≺ s. (b) Wt♯ is the normal subgroup of Ws♯ consisting of all elements that fix every [L♯ , σ ♯ ]L♯ with [L♯ , σ ♯ ]G♯ ≺ s. In particular it contains Ws . (c) There exist subgroups R♯s ⊂ Ws♯ , Rt♯ ⊂ Wt♯ and R♯ω ⊂ Wω♯ such that Ws♯ = Ws ⋊ R♯s , Wt♯ = Ws ⋊ Rt♯ and Wω♯ = Wω ⋊ R♯ω . Proof. (a) We may use the conditions 1.1. Then Wω = Ws . For any σ ♯ as above the root systems Rσ♯ and Rω are equal by [ChLi, Lemma 3.4.1]. Consequently Ws = Wω = W (Rω ) = W (Rσ♯ ) ⊂ Wσ♯ , showing that Ws fixes σ ♯ and [L♯ , σ ♯ ]L♯ . (b) As we observed above, we can arrange that every σ ♯ is a subquotient of ResL L♯ (ω). Since all constituents of this representation are associate under L and W (G, L) acts trivially on L/L♯ , Wt♯ fixes every possible [L♯ , σ ♯ ]L♯ . This gives the description of Wt♯ . Because all the [L♯ , σ ♯ ]L♯ together form a Ws♯ -space, Wt♯ is normal in that group. (c) In the special case G♯ = SLn (F ), this was proven for Wt♯ in [GoRo2, Proposition 2.3]. Our proof is a generalization Q of that in [GoRo2]. Recall the description of M = i Mi and Ws from equations (7) and (8). We note that P ∩ M is a parabolic subgroup of M containing L, and that the group HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 13 W (M, L) = Ws acts simply transitively on the collection of such parabolic subgroups. This implies that R♯s := Ws♯ ∩ NG (P ∩ M )/L (28) is a complement to Ws in Ws♯ . For Wt♯ (27) shows that we may take (29) Rt♯ := Wt♯ ∩ NG (P ∩ M )/L. Similarly, for Wω♯ (23) leads us to R♯ω := Wω♯ ∩ NG (P ∩ M )/L. (30)  As analogues of X L (ω), X G (IPG (ω)) and Stab(ω) for s = [L, ω]G we introduce (31) X L (s) = {γ ∈ Irr(L/L♯ Z(G)) | γ ⊗ ω ∈ [L, ω]L }, (32) X G (s) = {γ ∈ Irr(G/G♯ Z(G)) | γ ⊗ IPG (ω) ∈ s}, (33) Stab(s) = {(w, γ) ∈ W (G, L) × Irr(L/L♯ Z(G)) | w(γ ⊗ ω) ∈ [L, ω]L }. Notice that Stab(s) contains Stab(ω ⊗ χ) for every χ ∈ Xnr (L/L♯ ). It is easy to see that Ws × X L (s) is a normal subgroup of Stab(s) and that there are short exact sequences (34) 1 → X L (s) → Stab(s) → Ws♯ → 1, (35) 1 → X L (s) × Ws → Stab(s) → Ws♯ /Ws ∼ = R♯s → 1. Furthermore we define Stab(s, P ∩ M ) = {(w, γ) ∈ Stab(s) | w ∈ NG (P ∩ M )/L}. Lemma 2.4. (a) Stab(s) = Stab(s, P ∩ M ) ⋉ Ws . (b) The projection of Stab(s) on the second coordinate gives a group isomorphism Stab(s, P ∩ M ) ∼ = Stab(s)/Ws → X G (s). (c) The groups X L (s), X G (s) and Stab(s) are finite. (d) There are natural isomorphisms X G (s)/X L (s) ∼ = R♯ . = Stab(s, P ∩ M )/X L (s) ∼ s Proof. (a) This can be shown in the same way as Lemma 2.3.c. (b) If (w, γ) ∈ Stab(s), then γ ⊗ I G (ω) ∼ = γ ⊗ I G (w · ω) ∼ = I G (γ ⊗ w · ω) ∼ = I G (w(γ ⊗ ω)) ∈ s, P X G (s). P P X G (s), P IPG (ω so γ ∈ Conversely, if γ ∈ then ⊗ γ) ∈ s. Hence ω ⊗ γ ∈ w−1 · [L, ω]L = [L, w−1 · ω]L for some w ∈ W (G, L), and (w, γ) ∈ Stab(s). As W (G, L) and Irr(L/L♯ ) commute, the projection map Stab(s) → X G (s) is a group homomorphism. In view of (11), we may assume that ω is such that Ws = Wω . Then the kernel of this group homomorphism is Wω = Ws . (c) Suppose that ω ⊗ γ ∼ = ω ⊗ χ for some χ ∈ Xnr (L). Then χ is trivial on L♯ Z(G), md so χ = 1. Hence there are only finitely many possibilities for χ. We already know from Proposition 2.1 and (12) that X L (ω) finite, so we can conclude that X L (s) is finite. If (w, γ), (w, γ ′ ) ∈ Stab(s), then (w, γ)−1 (w, γ ′ ) = γ −1 γ ′ ∈ X L (s). As W (G, L) and X L (s) are finite, this shows that Stab(s) is also finite. Now X G (s) is finite by 14 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD part (b). (d) This follows from (35) and part (b).  2.3. The intermediate group. For some calculations it is beneficial to do the restriction of representations from G to G♯ in two steps, via the intermediate group G♯ Z(G). This is a central extension of G♯ , so EndG♯ (π) = EndG♯ Z(G) (π) (36) for all representations π of G or G♯ Z(G) that admit a central character. In particular G♯ Z(G) ResG♯ (37) preserves irreducibility of representations. The centre of G is Z(G) = G ∩ Z(Mm (D)) = G ∩ F · Im = F × Im . Recall that dimF (D) = d2 . Since Nrd(zIm ) = z md for z ∈ F × , Nrd(Z(G)) = F ×md , the group of md-th powers in F × . Hence G/G♯ Z(G) is an abelian group and all its elements have order dividing md. In case char(F ) is positive and divides md, G♯ Z(G) is closed but not open in G. Otherwise it is closed, open and of finite index in G. However, G♯ Z(G) is never Zariski-closed in G. The intersection of G♯ and Z(G) is the finite group {zIm | z ∈ F × , z md = 1}, so (38) G♯ Z(G) ∼ = (G♯ × Z(G))/{(zIm , z −1 ) | z ∈ F × , z md = 1}. As G♯ × Z(G) is a connected reductive algebraic group over F , this shows that G♯ Z(G) is one as well. But this algebraic structure is not induced from the enveloping group G. The inflation functor Rep(G♯ Z(G)) → Rep(G♯ × Z(G)) identifies Rep(G♯ Z(G)) with {π ∈ Rep(G♯ × Z(G)) | π(zIm , z −1 ) = 1 ∀z ∈ F × with z md = 1}. Lemma 2.5. (a) Every irreducible G♯ -representation can be lifted to an irreducible representation of G♯ Z(G). (b) All fibers of G♯ Z(G) ResG♯ : Irr(G♯ Z(G)) → Irr(G♯ ) are homeomorphic to Irr(F ×md ). Proof. (a) Any π ♯ ∈ Irr(G♯ ) determines a character χmd of the central subgroup {z ∈ F × | z md = 1}. Since there are only finitely many md-th roots of unity in F, χmd can be lifted to a character χ of F × . Then π ♯ ⊗ χ is a representation of G♯ × Z(G) that descends to G♯ Z(G). (b) This follows from the proof of part (a) and the short exact sequence (39) Nrd 1 → G♯ → G♯ Z(G) −−→ F × → 1.  More explicitly, χ ∈ Irr(F ×md ) acts on Irr(G♯ Z(G)) by retraction to χ̄ ∈ Irr(G♯ Z(G)) and tensoring representations of G♯ Z(G) with χ̄. It follows that the ♯ preimage of Irrt (G♯ ) in Irr(G♯ Z(G)) consists of countably many Bernstein components Irrt (G♯ Z(G)), each one homeomorphic to ♯ ♯ ♯ Xnr (F ×md ) × Irrt (G♯ ) ∼ = Xnr (G♯ Z(G)) × Irrt (G♯ ) ∼ = C× × Irrt (G♯ ). HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 15 Two such components differ from each other by a ramified character of F ×md , or equivalently by a character of Nrd(o× F · 1m ). In comparison, every Bernstein component Irrt (G♯ Z(G)) projects onto a single ♯ Bernstein component for G♯ , say Irrt (G♯ ). All the fibers of (40) G♯ Z(G) ResG♯ ♯ : Irrt (G♯ Z(G)) → Irrt (G♯ ) are homeomorphic to Xnr (Z(G)) = Xnr (F ×md ) ∼ = C× . In particular (41) Tt♯ = Tt /Xnr (Nrd(Z(G))). Lemma 2.6. The finite groups associated to t and t♯ are equal: Wt = Wt♯ . Proof. As we observed above, t♯ is the only Bernstein component involved in the ♯ restriction of Irrt (G♯ Z(G)) to G♯ . Hence W t ⊂ W t . Conversely, if w ∈ W (L♯ ) ′ stabilizes t♯ , then it stabilizes the set of Bernstein components Irrt (G♯ Z(G)) which ♯ project onto Irrt (G♯ ). But any such t′ differs from t only by a ramified character of Z(G). Since conjugation by elements of NG (L) does not affect Z(G), w(t) cannot be another t′ , and so w ∈ Wt .  G♯ Z(G) , so we can focus on ResG The above provides a complete picture of ResG♯ G♯ Z(G) . ♯ Although [G : G Z(G)] is sometimes infinite (e.g. if char(F ) divides md), only finitely many characters of G/G♯ Z(G) occur in relation to a fixed Bernstein component. This follows from Lemma 2.4.c and makes it possible to treat G♯ Z(G) ⊂ G as a group extension of finite degree. Given an inertial equivalence class s = [L, ω]G , we define Irrs (G♯ Z(G)) as the set of all elements of Irr(G♯ Z(G)) that can be obtained as a subquotient of ResG G♯ Z(G) (π) s s ♯ for some π ∈ Irr (G). We also define Rep (G Z(G)) as the collection of G♯ Z(G)representations all whose irreducible subquotients lie in Irrs (G♯ Z(G)). It follows from Lemma 2.2 and the above that Irrs (G♯ Z(G)) is a union of finitely many Bernstein components t for G♯ Z(G). We denote this relation between s and t by t ≺ s. Thus [ (42) Irrs (G♯ Z(G)) = Irrt (G♯ Z(G)). t≺s All the reducibility of G-representations caused by restricting them to G♯ can already be observed by restricting them to G♯ Z(G). In view of (36) and Lemma 2.6, all our ♯ ♯ ♯ results on ResG G♯ remain valid if we replace everywhere G by G Z(G) and L by L♯ Z(G). In view of (15), intertwining operators associated to Stab(s) span EndG♯ Z(G) (IPG (ω ⊗ χ)) whenever χ ∈ Xnr (L) is unitary. With results of HarishChandra we will show that even more is true. Let (43) J(w, IPG (ω ⊗ χ))) ∈ HomG (IPG (ω ⊗ χ), IPG (w(ω ⊗ χ))) be the intertwining operator constructed in [Sil, §5.5.1] and [Wal, §V.3]. We recall that it is rational as a function of χ ∈ Xnr (L) and that it is regular and invertible if χ is unitary. In contrast to the intertwining operators below, (43) can be normalized in a canonical way. For (w, γ) ∈ Stab(s, P ∩ M ) there exists a χ′ ∈ Xnr (L), unique up to Xnr (L, ω), such that w(ω ⊗ χγ) ∼ = ω ⊗ χ′ . Choose a nonzero (44) J(γ, ω ⊗ χ) ∈ HomL (ω ⊗ χ, w−1 (ω ⊗ χ′ γ −1 )). 16 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD In view of Lemma 2.4.b γ determines w, so this is unambigous and determines J(γ, ω ⊗ χ) up to a scalar. For unramified γ we have χ′ = χγ, but nevertheless J(γ, ω ⊗ χ) need not be a scalar multiple of identity. The reason lies in the difference between Ts and Xnr (L), we refer [Wal, §V] for more background. Parabolic induction produces (45) J(γ, IPG (ω ⊗ χ)) := IPG (J(γ, ω ⊗ χ)) ∈ HomG (IPG (ω ⊗ χ), IPG (w−1 (ω ⊗ χ′ γ −1 ))). For w′ (w, γ) ∈ Stab(s) with w′ ∈ Ws and (w, γ) ∈ Stab(s, P ∩ M ) we define (46) J(w′ (w, γ), IPG (ω ⊗ χ)) := J(w′ , IPG (ω ⊗ χ′ γ −1 )) ◦ J(w, IPG (w−1 (ω ⊗ χ′ γ −1 ))) ◦ J(γ, IPG (ω ⊗ χ)). By construction this lies both in HomG (IPG (ω ⊗ χ), IPG (w′ (ω ⊗ χ′ γ −1 ))) and in HomG♯ Z(G) (IPG (ω ⊗ χ), IPG (w′ (ω ⊗ χ′ ))). We remark that the map from Stab(s) to intertwining operators (46) is not always multiplicative, some 2-cocycle with values in C× might be involved. However, (47) Ws ∋ w′ 7→ {J(w′ , IPG (ω ⊗ χ)) | χ ∈ Xnr (L)} is a group homomorphism. The following result is the main justification for introducing Stab(s). Theorem 2.7. (a) For χ1 , χ2 ∈ Xnr (L) the following are equivalent: G G G (i) ResG G♯ Z(G) (IP (ω ⊗ χ1 )) and ResG♯ Z(G) (IP (ω ⊗ χ2 )) have a common irreducible subquotient; G G G (ii) ResG G♯ Z(G) (IP (ω⊗χ1 )) and ResG♯ Z(G) (IP (ω⊗χ2 )) have the same irreducible constituents, counted with multiplicity; (iii) ω ⊗ χ1 and ω ⊗ χ2 belong to the same Stab(s)-orbit. (b) If χ1 and χ2 are unitary, then HomG♯ Z(G) (IPG (ω ⊗ χ1 ), IPG (ω ⊗ χ2 )) is spanned by intertwining operators J((w, γ), IPG (ω ⊗ χ1 )) with (w, γ) ∈ Stab(s) and w(ω ⊗ χ1 γ) ∼ = ω ⊗ χ2 . Proof. First we assume that χ1 and χ2 are unitary. By Harish-Chandra’s Plancherel isomorphism [Wal] and the commuting algebra theorem [Sil, Theorem 5.5.3.2], the theorem is true for G, with Ws instead of Stab(s). More generally, for any tempered ρ1 , ρ2 ∈ Irr(L), HomG (IPG (ρ1 ), IPG (ρ2 )) is spanned by intertwining operators J(w, IPG (ρ1 )) with w ∈ W (G, L) and w · ρ1 ∼ = ρ2 . G For π1 , π2 ∈ Irr(G) Proposition 2.1.b says that ResG G♯ Z(G) (π1 ) and ResG♯ Z(G) (π2 ) are isomorphic if π2 ∼ = π1 ⊗ γ for some γ ∈ Irr(G/G♯ Z(G)), and have no common irreducible subquotients otherwise. Together with (15) this implies that HomG♯ Z(G) (IPG (ω ⊗ χ1 ), IPG (ω ⊗ χ2 )) ∼ is spanned by intertwining operators J((w, γ), IPG (ω ⊗ χ1 )) with w(ω ⊗ χ1 γ) = ω ⊗ χ2 . Such pairs (w, γ) automatically belong to Stab(s). Since both factors of J((w, γ), IPG (ω ⊗ χ1 )) are bijective, the equivalence of (i), (ii) and (iii) follows. This proves (b) and (a) in the unitary case. Now we allow χ1 and χ2 to be non-unitary. Assume (i). From Proposition 2.1.b we obtain a γ ∈ Irr(G/G♯ Z(G)) such that IPG (ω ⊗ χ1 γ) and IPG (ω ⊗ χ2 ) have a common irreducible quotient. The theory of the Bernstein centre for G [BeDe] HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 17 implies that ω ⊗ χ1 γ and ω ⊗ χ2 are isomorphic via an element w ∈ W (G, L). Then (w, γ) ∈ Stab(s), so (iii) holds. Suppose now that w(ω ⊗ χ1 γ) ∼ = ω ⊗ χ2 for some (w, γ) ∈ Stab(s) and consider the map (48) H(G♯ Z(G)) × Xnr (L) → C : (f, χ) 7→ tr(f, IPG (ω ⊗ χ)) − tr(f, IPG (w(ω ⊗ γχ))). G It is well-defined since ResG G♯ IP (ω ⊗ χ) has finite length, by Proposition 2.1.c. By what we proved above, the value is 0 whenever χ is unitary. But for a fixed f ∈ H(G) this is a rational function of χ ∈ Xnr (L), and the unitary characters are Zariskidense in Xnr (L). Therefore (48) is identically 0, which shows that (IPG (ω ⊗ χ)) and IPG (w(ω ⊗ γχ)) have the same trace. By Proposition 2.1.c these G♯ -representations have finite length, so by [Cas, 2.3.3] their irreducible constituents (and multiplicities) are determined by their traces. Thus (iii) implies (ii), which obviously implies (i).  3. Morita equivalences Let s = [L, ω]G . We want to analyse the two-sided ideal H(G♯ Z(G))s of H(G♯ Z(G)) associated to the category of representations Reps (G♯ Z(G)) introduced in Subsection 2.3. In this section we will transform these algebras to more manageable forms by means of Morita equivalences. We note that by (42) we can regard H(G♯ Z(G))s as a finite direct sum of ideals associated to one Bernstein component: M (49) H(G♯ Z(G))s = H(G♯ Z(G))t . t≺s Recall that the abelian group Irr(G/G♯ ) acts on H(G) by (χ · f )(g) = χ(g)f (g). We also introduce an alternative action of γ ∈ Irr(G/G♯ ) on H(G) (and on similar algebras): αγ (f ) = γ −1 · f. (50) Obviously these two actions have the same invariants. An advantage of the latter lies in the induced action on representations: αγ (π) = π ◦ α−1 γ = π ⊗ γ. Suppose for the moment that the characteristic of F does not divide md, so that G/G♯ Z(G) is finite. Then there are canonical isomorphisms M (51) H(G♯ Z(G))s ∼ = H(G♯ Z(G)) s∈B(G)/∼ M ♯ G ∼ (H(G)s )X (s) , = H(G)Irr(G/G Z(G)) ∼ = s∈B(G)/∼ s′ where s ∼ if and only if they differ by a character of G/G♯ Z(G). Unfortunately this is not true if char(F ) does divide md. In that case there are no nonzero Irr(G/G♯ Z(G))-invariant elements in H(G), because such elements could not be locally constant as functions on G. In Subsection 3.3 we will return to this point and show that it remains valid as a Morita equivalence. Throughout this section will assume from now on that the conditions 1.1 are in force. 18 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD 3.1. Construction of a particular idempotent. We would like to find a type which behaves well under restriction from G to G♯ . As this is rather complictated, we start with a simpler goal: an idempotent in H(M ) which is suitable for restriction from M to M ♯ . Recall from [SéSt2] that there exists an s-type (KG , λG ), and that it can be constructed as a cover of a type (K, λ) for sM = [L, ω]M . We refer to [BuKu3, Section 8] for the notion of a cover of a type. For the moment, it suffices to know that K = KG ∩ M and that λ is the restriction of λG to K. In fact, in [Séc] (K, λ) is exhibited as a cover of a type (KL , λL ) for sL = [L, ω]L . From (8) we know that NG ([L, ω]L ) ⊂ M . In this situation [BuKu3, Theorem 12.1] says that there is an algebra isomorphism (52) eλ H(G)eλ ∼ = eλ H(M )eλ G G and that the normalized parabolic induction functor IPGM : RepsM (M ) → Reps (G) is an equivalence of categories. By Conditions 1.1 and (7) we may assume that (K, λ) factors as Y Y K= (K ∩ Mi ) =: Ki , i (53) O O i
λi , (λ, Vλ ) = Vλi . i i Moreover we may assume that, whenever mi = mj and ωi and ωj differ only by a character of Li /L♯i , Ki = Kj and λi and λj also differ only by a character of Ki /(Ki ∩ L♯i ). We note that these assumptions imply that R♯s normalizes K. By respectively (52), (6) and (53) there are isomorphisms O (54) eλG H(G)eλG ∼ H(Mi , λi ) ⊗C EndC (Vλi ). = H(M, λ) ⊗C EndC (Vλ ) ∼ = i We need more specific information about the type (K, λ) in M . To study this and the related types (K, w(λ) ⊗ γ) we will make ample use of the theory developed by Sécherre and Stevens [Séc, SéSt1, SéSt2]. In [Séc] (K, λ) arises as a cover of a [L, ω]L -type (KL , λL ). In particular λ is trivial on both K ∩ N and K ∩ N , where N and N are the unipotent radicals of P L ∩ M and of the opposite parabolic subgroup of M , and λL is the restriction of λ to KL = K ∩ L. Proposition 3.1. We can choose the sM -type (K, λ) such that, for all (w, γ) ∈ Stab(s), (K, w(λ) ⊗ γ) is conjugate to (K, λ) by an element cγ ∈ L. Moreover cγ Z(L) lies in a compact subgroup of L/Z(L) and we can arrange that cγ depends only on the isomorphism class of w(λ) ⊗ γ ∈ Irr(K). Remark. For GLn (F ) very similar results were proven in [GoRo1, §4.2], using [BuKu4]. Proof. By definition (K, λ) and (K, w(λ) ⊗ γ) are both types for [L, ω]M . By Conditions 1.1 and (53) they differ only by a character of M/M ♯ Z(G) ∼ = G/G♯ Z(G), L L which automatically lies in X (w(s) ⊗ γ) = X (s). Hence it suffices to prove the proposition in the case w = 1, γ ∈ X L (s). This setup implies that we consider (w, γ) only modulo isomorphism of the representations w(λ) ⊗ γ. HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 19 Q In view of the factorizations M = i Mi and (53), we can treat the various i’s separately. Thus we may assume that Mi = G. To get in line with [Séc], we temporarily change the notation to G = GLmi e (D), L = GLmi (D)e , JP = Ki and ω = ωi⊗e . We need a type (JP , λP ) for [L, ω]G with suitable properties. We will use the one constructed in [Séc] as a cover of a simple type (Jie , λ⊗e i ) for [L, ω]L . Analogously there is a cover (JP , λP ⊗ γ) of the [L, ω]L -type e ⊗e (Jie , λ⊗e i ⊗ γ) = (Ji , (λi ⊗ γ) ). In these constructions (Ji , λi ) and (Ji , λi ⊗ γ) are two maximal simple types for the supercuspidal inertial equivalence class [GLmi (D), ωi ]GLmi (D) . According to [SéSt2, Corollary 7.3] they are conjugate, say by ci ∈ GLmi (D). Then (Jie , λ⊗e i ) and (Jie , λ⊗e ⊗ γ) are conjugate by i cγ,i := diag(ci , ci , . . . , ci ) ∈ L. Recall that P is the parabolic subgroup of G generated by L and the upper triangular matrices. Let N be the unipotent radical of P and N the unipotent radical of the parabolic subgroup opposite to P . The group JP constructed in [Séc, §5.2] and [SéSt1, §5.5] admits an Iwahori decomposition (55) JP = (JP ∩ N )(JP ∩ L)(JP ∩ N ) = (H 1 ∩ N ) Jie (J ∩ N ). Let us elaborate a little on the subgroups H 1 and JP ⊂ J of G. In [SéSt1] a certain stratum [C, n0 , 0, β] is associated to (GLmi (D), ωi ), which gives rise to compact open subgroups Hi1 and Ji of GLmi (D). From this stratum Sécherre [Séc, 5.2.2] defines another stratum [A, n, 0, β] associated to (L, ωi⊗e ), which in the same way produces H 1 and J. The procedure entails that H 1 and J can be obtained by putting together copies of Hi1 , Ji and their radicals in block matrix form. The proofs of [SéSt2, Theorem 7.2 and Corollary 7.3] show that we can take ci such that it normalizes Ji and Hi1 . Then it follows from the explicit relation between the above two strata that cγ,i normalizes J and H 1 . Notice also that cγ,i normalizes N and N , because it lies in L. Hence cγ,i normalizes JP and its decomposition (55). By definition [Séc, 5.2.3] the representation λP of JP is trivial on JP ∩ N and on JP ∩ N , whereas its restriction to JP ∩ L equals λ⊗e i . As ci conjugates λi to λi ⊗ γ, we deduce that cγ,i conjugates (JP , λP ) to (JP , λP ⊗ γ). Q To get back to the general case we recall that M = i Mi and we put Y Y (56) cγ := cγ,i = diag(ci , ci , . . . , ci ). i i It remains to see that cγ becomes a compact element in L/Z(L). Since Ji is open and compact, its fixed points in the semisimple Bruhat–Tits building B(GLmi (D)) form a nonempty bounded subset. Then ci stabilizes this subset, so by the Bruhat–Tits fixed point theorem ci fixes some point xi ∈ B(GLmi (D)). But the stabilizer of xi is a compaact modulo centre subgroup, so in particular ci is compact modulo centre. Therefore the image of cγ in L/Z(L) is a compact element.  In the above proof it is also possible to replace (JP , λP ) by a sound simple type in the sense of [SéSt2]. Indeed, the group J from [Séc, §5] is generated by JP and J ∩ N , so it is also normalized by cγ,i . By [Séc, Proposition 5.4] (J, IndJJP (λP )) and (J, IndJJP (λP ⊗ γ)) 20 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD are sound simple types. The above proof also shows that they are conjugate by cγ,i . As noted in the proof of [Séc, Proposition 5.5], there is a canonical support preserving algebra isomorphism ∼ H(Mi , IndJ (λP )). (57) H(Mi , λP ) = JP In particular the structure theory of the Hecke algebras in [Séc] also applies to our types (K, λ). We write \ L1 := ker χ. G1 χ∈Xnr (L) o× F } is the group generated by all compact Notice that = {g ∈ G | Nrd(g) ∈ subgroups of G. Hence L1 is the group generated by all compact subgroups of L. We fix a choice of elements cγ ∈ L as in Proposition 3.1, such that cγ ∈ L1 whenever possible. This determines subgroups (58) X L (s)1 := {γ ∈ X L (s) | cγ ∈ L1 }, 1 Stab(s, P ∩ M ) := {(w, γ) ∈ Stab(s, P ∩ M ) | cγ ∈ L1 }. Their relevance is that the cγ ∈ L1 can be used to construct larger sM -types from (K, λ), whereas the cγ with γ ∈ X L (s) \ X L (s) are unsuitable for that purpose. We remark that in the split case G = GLn (F ) it is known from [BuKu2, Proposition 2.2] that one can find cγ ∈ L1 for all γ ∈ X L (s). Consider the group Stab(s, λ) = {(w, γ) ∈ Stab(s, P ∩ M ) | w(λ) ⊗ γ ∼ = λ as K-representations}. The elements of this group are precisely the (w, γ) ∈ Stab(s, P ∩ M ) for which ew(λ)⊗γ = eλ . Lemma 3.2. Projection on the first coordinate gives a short exact sequence 1 → X L (s) ∩ Stab(s, λ) → Stab(s, λ) → R♯s → 1. The inclusion X L (s) → Stab(s, P ∩ M ) induces a group isomorphism X L (s)/(X L (s) ∩ Stab(s, λ)) → Stab(s, P ∩ M )/Stab(s, λ). Proof. Let (w, γ) ∈ Stab(s, P ∩ M ). Then w(λ)⊗ γ ∼ = λ⊗ γ ′ for some γ ′ ∈ X L (s) and ′ (w, γ) ∈ Stab(s, λ) if and only if γ ∈ Stab(s, λ). Hence all the fibers of Stab(s, λ) → R♯s have the same cardinality, namely |X L (s) ∩ Stab(s, λ)|. The required short exact sequence follows. The asserted isomorphism of groups is a direct consequence thereof.  Motivated by Lemma 3.2 we abbreviate X L (s, λ) := X L (s) ∩ Stab(s, λ), (59) X L (s/λ) := X L (s)/X L (s, λ), X L (s/λ)1 := X L (s)1 /X L (s, λ). The latter two groups are isomorphic to respectively Stab(s, P ∩ M )/Stab(s, λ) and Stab(s, P ∩ M )1 /Stab(s, λ). P By Lemma 3.2 the element γ∈X L (s/λ) eλ⊗γ ∈ H(K) is well-defined and idempotent. Clearly this element is invariant under X L (s), which makes it more suitable to study the restriction of Reps (G) to G♯ . However, in some cases this idempotent sees only HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 21 a too small part of a G-representation. This will become apparant in the proof of Proposition 3.8.d. We need to replace it by a larger idempotent, for which we use the following lemma. Lemma 3.3. There exist a subgroup Hλ ⊂ L and a subset [L/Hλ ] ⊂ L such that: (a) [L/Hλ ] is a set of representatives for L/Hλ , where Hλ ⊂ L is normal, of finite index and contains L♯ Z(L). (b) Every element of [L/Hλ ] commutes with Ws♯ and has finite order in L/Z(L). (c) For every χ ∈ Xnr (L) the space X X aeλL ⊗γ a−1 Vω⊗χ L a∈[L/Hλ ] γ∈X (s) L♯ -isotypical intersects every component of Vω⊗χ nontrivially. sM (d) For every (π, Vπ ) ∈ Irr (M ) the space X X aeλ⊗γ a−1 Vπ L a∈[L/Hλ ] γ∈X (s) intersects every M ♯ -isotypical component of Vπ nontrivially. Proof. (a) First we identify the group Hλ . Recall the operators I(γ, ω) ∈ HomL (ω ⊗ γ, ω) from (13), with γ ∈ X L (s, λ) ∩ X L (ω). Like in (16) and [HiSa, Corollary 2.10], these provide a decomposition of L♯ -representations M ω= HomC[X L (s,λ)∩X L (ω),κω ] (ρ, ω) ⊗ ρ. ρ∈Irr(C[X L (s,λ)∩X L (ω),κω ]) Let us abbreviate it to (60) Vω = M ρ Vω,ρ . It follows from Proposition 2.1 and (16) that all the summands Vω,ρ are L-conjugate and that StabL (Vω,ρ ) is a finite index normal subgroup which contains L♯ Z(L). This leads to a bijection (61) Irr(C[X L (s, λ) ∩ X L (ω), κω ]) ←→ L/StabL (Vω,ρ ). We claim that X (62) γ∈X L (s) eλL ⊗γ Vω is an irreducible representation of a subgroup N ⊂ L that normalizes KL . From [Séc, Théorème 4.6] it is known that eλ ⊗γ H(L)eλ ⊗γ ∼ = O(Ts ) ⊗ EndC (Vλ ⊗γ ), L L L which implies that every eλL ⊗γ Vω is irreducible as a representation of KL Z(L). These representations, with γ ∈ X L (s/λ) are inequivalent and permuted transitively by the elements cγ from Proposition 3.1. Hence (62) is irreducible as a representation of the group N generated by KL Z(L) and the cγ . Suppose now that (62) intersects both Vω,ρ1 and Vω,ρ2 nontrivially. By the above claim, N contains an element that maps Vω,ρ1 to Vω,ρ2 . It follows that, under the bijection (61), the set of ρ’s such that Vω,ρ intersects (62) nontrivially corresponds to a subgroup of L/StabL (Vω,ρ ), say Hλ /StabL (Vω,ρ ). Because L/StabL (Vω,ρ ) was already finite and abelian, Hλ has the desired properties. We note that none of the above changes if we twist ω by an unramified character of L. 22 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Q (b) Recall that L = i GLmi (D)ei and that the reduced norm map D × → F × is surjective. It provides a group isomorphism L/Hλ → F × /Nrd(Hλ ), and Nrd(Hλ ) contains Nrd(Z(L)) = F ×e where e is the greatest common divisor of the numbers mi . We can choose explicit representatives for L/Hλ . It suffices to use elements a whose components in ai ∈ GLmi (D) are powers of some element of the form   0 1 0 ··· 0 0 0 1 · · · 0    .. ..  . .  ∈ GLmi (D),   0 0 ··· 0 1 di 0 ··· 0 0 that is, the matrix of the permutation (1 mi (mi − 1) · · · 32), with one entry replaced by an element di ∈ D × . In this way we assure that a has finite order in L/Z(L), at most ed. Q If two factors Li = GLmi (D) and Lj = GLmj (D) of L = i Lei i are conjugate via an element of Ws♯ , then mi = mj and the corresponding supercuspidal representations ωi and ωj differ only by a character of GLmi (D), say η. As in the proof of Proposition 3.1, let (Ji , λi ) be a simple type for (Li , ωi ). As in (53) we use the type (Ji , λ ⊗ η) for (Lj , ωj ). Given γ ∈ X L (s, λ) ∩ X L (ω), we can factor Y I(γ, ωi )⊕ei , I(γ, ω) = i with I(γ, ωi ) ∈ HomLi (ωi ⊗ γ, ωi ). Here can simply take I(γ, ωj ) = I(γ, ωi ). Then the decomposition of Vωj in isotypical subspaces Vωi ,ρ for C[X L (s, λ) ∩ X L (ω), κω ], like (60) is the same as that of Vωi , and M i Vω,ρ = Vω⊕e . i ,ρ i Suppose now that a component ai of a as above maps Vωi ,ρ to Vωi ,ρ′ . Then ai also Q maps Vωj ,ρ to Vωj ,ρ′ , so we may take aj = ai . With this construction a = i ai⊕ei commutes with Ws♯ . We fix such a set of representatives a and denote it by (63) [L/Hλ ] = {al | l ∈ L/Hλ }. (c) Let χ ∈ Xnr (L). By construction X X (64) a∈[L/Hλ ] γ∈X L (s) aeλL ⊗γ a−1 Vω⊗χ intersects Vω⊗χ,ρ nontrivially for every ρ ∈ Irr(C[X L (s, λ) ∩ X L (ω), κω ]). All the idempotents aeλL ⊗γ a−1 are invariant under X L (s, λ) ∩ X L (ω) because eλL is. Hence aeλL ⊗γ a−1 Vω⊗χ is nonzero for at least one of these idempotents. The action of X L (ω)/X L (ω) ∩ Stab(s, λ) permutes the idempotents aeλL ⊗γ a−1 faithfully, so by Frobenius reciprocity the space X aeλL ⊗γ a−1 Vω⊗χ L γ∈X (s) HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 23 contains all irreducible representations of C[X L (ω ⊗χ), κω⊗χ ] that contain ρ. Therefore (64) contains all irreducible representations of C[X L (ω ⊗ χ), κω⊗χ ]. With (16) (for L) this says that (64) intersects every L♯ -isotypical component of Vω⊗χ nontrivially. (d) Let π ∈ IrrsM (M ) and choose χ ∈ Xnr (L) such that π is a subquotient of IPM∩M (ω ⊗ χ). Lemma 2.4.d. in combination with the equality W (M, L) = Ws shows that X M (π) ⊂ X L (ω ⊗ χ) = X L (ω). Since (K, λ ⊗ γ) is a type, aeλ⊗γ a−1 Vπ 6= 0 for all possible a, γ. The group X L (s, λ) ∩ X M (π) effects a decomposition of the M ♯ -representation Vπ by means of the operators IPM∩M (γ, ω ⊗ χ). Analogous to (60) we write it as M (65) Vπ = Vπ,ρ . L M ρ∈Irr(C[X (s,λ)∩X The construction of Hλ entails that X X a∈[L/Hλ ] γ∈X L (s) (π),κω ]) aeλ⊗γ a−1 Vπ intersects every summand Vπ,ρ of (65) nontrivially. Now the same argument as for part (c) shows that this space intersects every M ♯ -isotypical component of Vπ nontrivially.  With (59) and Lemma 3.3 we construct some additonal idempotents: X eµL := eλL ⊗γ ∈ H(KL ), γ∈X L (s/λ) X esL := aeµL a−1 ∈ H(L), a∈[L/Hλ ] (66) X eµ := eλ⊗γ ∈ H(K), γ∈X L (s/λ) X esM := aeµ a−1 ∈ H(M ). a∈[L/Hλ ] Lemma 3.4. The four elements in (66) are idempotent and Stab(s, P ∩M )-invariant. Furthermore eµL , esL ∈ H(L)sL and eµ , esM ∈ H(M )sM . Proof. We only write down the proof for the last two elements, the argument for the first two is analogous. We already observed that the different idempotents eλ⊗γ are orthogonal, so that −1 ′ their sum eµ is again idempotent. We claim that e = al eλ⊗γ a−1 l and e = al′ eλ⊗γ ′ al′ ′ ′ are orthogonal unless l = l and γ = γ . By construction, the images of e and e′ in EndC (VI M ω ) are orthogonal. This P ∩M remains true if we twist ω by an unramified character χ ∈ Xnr (L). But the M representations IPM∩M (ω⊗χ) together generate the entire category RepsM (M ). Hence ee′ = e′ e = 0 on every representation in RepsM (M ). Since eλ ∈ H(M )sM and that algebra is stable under conjugation with elements of M and under Stab(s) by (69), all the aeλ⊗γ a−1 lie in H(M )sM . Thus e, e′ ∈ H(M )sM , and we can conclude that they are indeed orthogonal. This implies that esM ∈ H(M )sM is idempotent. 24 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Since eλ⊗γ is invariant under X L (s, λ), so is eµ . The action of X L (s) commutes with conjugation by any element of M , hence the sum over γ ∈ X L (s/λ) in the definition of eµ makes it X L (s)-invariant. By (53) and the last part of Proposition 3.1, eµ is invariant under Stab(s, P ∩ M ) (but not necessarily under Ws ). By Lemma 3.3.b this remains the case after conjugation by any a ∈ [L/Hλ ]. Hence aeµ a−1 and esM are also invariant under Stab(s, P ∩ M ).  We can interpret the group L/Hλ in a different way. Define (67) Vµ := eµ Vω , X L (ω, Vµ ) := {γ ∈ X L (ω) | I(γ, ω)|Vµ ∈ C× idVµ }. Lemma 3.5. There is a group isomorphism L/Hλ ∼ = Irr(X L (ω, Vµ )). Proof. We use the notation from the proof of Lemma 3.3. Consider the twisted group algebra (68) C[X L (s, λ) ∩ X L (ω), κω ]. We noticed in (60) and (61) that all its irreducible representations have the same dimension, say δ. Let C be the subgroup of X L (s, λ) ∩ X L (ω) that consists of all elements γ for which I(γ, ω) acts as a scalar operator on Vω,ρ . Since all the Vω,ρ are L-conjugate, this does not depend on ρ. As the dimension of (68) equals |X L (s, λ) ∩ X L (ω)|, we find that [X L (s, λ) ∩ X L (ω) : C] = δ2 and |C| = [L : StabL (Vω,ρ )]. Since C acts on every Vω,ρ by a character, we can normalize the operators I(γ, ω) such that κω |C×C = 1. The subalgebra of (68) spanned by the I(γ, ω) with γ ∈ C has dimension |C|, so every character of C appears in Vω,ρ for precisely one ρ ∈ Irr(C[X L (s, λ) ∩ X L (ω), κω ]). Now we see from (61) that C = Irr(L/StabL (Vω,ρ )) and Irr(C) ∼ = L/StabL (Vω,ρ ). Under the this isomorphism the subgroup Hλ /StabL (Vω,ρ ) corresponds to the set of character of C that occur in Vµ . That set can also be written as Irr(C/X L (ω, Vµ )). Hence the quotient  L/Hλ = (L/StabL (Vω,ρ )) (Hλ /StabL (Vω,ρ )) is isomorphic to Irr(X L (ω, Vµ )).  3.2. Descent to a Levi subgroup. G Motivated by the isomorphisms (51) we focus on (H(G)s )X (s) . We would like to replace it by a Morita equivalent subalgebra of H(M )sM . However, the latter algebra is in general not stable under the action of X G (s). In fact, for (w, γ) ∈ Stab(s) we have (69) γ · H(M )sM = H(M )[L,ω⊗γ −1 ] M = H(M )[L,w·ω]M = H(M )w(sM ) . Let us regard R♯s as a group of permutation matrices in G. Then it acts on M by conjugation and we can form the crossed product H(M ⋊ R♯s ) = H(M ) ⋊ R♯s . HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 25 We define H(M ⋊ R♯s )s as the two-sided ideal of H(M ⋊ R♯s ) such that IndG (V ) ∈ Reps (G) for all V ∈ Mod(H(M ⋊ R♯s )s ). P M ⋊R♯ s Because = 0 for w ∈ R♯s \ {1}, we have M
w(sM ) H(M ⋊ R♯s )s = ⋊ R♯s . ♯ H(M ) H(M )sM H(M )w(sM ) (70) w∈Rs By (69) the algebra (70) is stable under X G (s). We extend the action α of X L (s) on H(M ) to Stab(s) by α(w,γ) (f ) := w(γ −1 · f )w−1 . (71) Given w ∈ R♯s , Lemma 2.4.d shows that there exists a γ ∈ Irr(L/L♯ Z(G)) such that (w, γ) ∈ Stab(s), and that γ is unique up to X L (s). Hence w 7→ α(w,γ) determines a L L group action of R♯s on (H(M ))X (s) . By (69) this action stabilizes (H(M )sM )X (s) . Using this action, we can rewrite the α-invariant subalgebra of (70) conveniently: Lemma 3.6. There is a canonical isomorphism
X G (s) X L (s) ∼ ⋊ R♯s . H(M ⋊ R♯s )s = (H(M )sM ) Proof. Using (70) and the fact that X G (s) fixes all elements of C[R♯s ], we can rewrite M
X G (s)
X G (s)/X L (s) w(sM ) X L (s) ∼ H(M ⋊ R♯s )s ) ⋊ R♯s . = ( ♯ H(M ) w∈Rs By Lemma 2.4.d this is M
R♯
♯ L sM X L (s) −1 Rs ∼ ) w2 = EndC (C[R♯s ]) ⊗ (H(M )sM )X (s) s . ♯ w1 (H(M ) w1 ,w2 ∈Rs In the right hand side the action of R♯s has become the regular representation on EndC (C[R♯s ]) tensored with the action α(w,γ) as in (71). By a folklore result (see [Sol1, Lemma A.3] for a proof) the right hand side is isomorphic to (H(M )sM )X R♯s . L (s) ⋊  In Proposition 3.8 we will show that the algebras from Lemma 3.6 are Morita G equivalent with (H(G)s )X (s) . We note that by Lemma 3.2 esM is Stab(s, P ∩ M )-invariant, so from (71) we obtain an action of Stab(s, P ∩ M ) on esM H(M ⋊ R♯s )s esM . Lemma 3.7. The following algebras are Morita equivalent: H(G)s , H(M )sM , H(M ⋊ R♯s )s , esM H(M )esM and esM H(M ⋊ R♯s )s esM . Proof. We will denote Morita equivalence with ∼M . The Morita equivalence of H(G)s and H(M )sM follows from the fact that NG (sL ) ⊂ M . It is given in one direction by (72) IPGM : Mod(H(M )sM ) = RepsM (M ) → Reps (G) = Mod(H(G)s ) and in the other direction by (73) prsM ◦ rPGM : Reps (G) → Reps (M ) → RepsM (M ), the normalized Jacquet restriction functor rPGM followed by projection on the factor RepsM (M ) of Reps (M ). The formula (70) shows that (74) H(M )sM ∼M H(M ⋊ R♯s )s , 26 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD the equivalence being given by H(M ⋊R♯ )s IndH(M )sMs ♯ ⋊Rs = IndM . M With the bimodules esM H(M )sM and H(M )sM esM we see that esM H(M )esM = esM H(M )sM esM ∼M H(M )sM esM H(M )sM . (75) Since (K, λ) is an sM -type, every module of H(M )sM is generated by its λ-isotypical vectors and a fortiori by the image of esM in such a module. Therefore H(M )sM esM H(M )sM = H(M )sM . The same argument, now additionally using (74), also shows that H(M ⋊ R♯s )s ∼M esM H(M ⋊ R♯s )s esM .  The above lemma serves mainly as preparation for some more involved Morita equivalences: Proposition 3.8. The following algebras are Morita equivalent to (H(G)s )X G X L (s) ⋊ R♯s ; (a) (H(M ⋊ R♯s )s )X (s) ∼ = (H(M )sM ) s (b) H(M ) M ⋊ Stab(s, P ∩ M ); (c) esM H(M )esM ⋊ Stab(s, P ∩ M ); L (d) (esM H(M )esM )X (s) ⋊ R♯s . G (s) : Proof. (a) The isomorphism between the two algebras is Lemma 3.6. Let U be the unipotent radical of P M . As discussed in [MeSo], there are natural isomorphisms I G (V ) ∼ V ∈ Rep(M ), = C ∞ (G/U ) ⊗H(M ) V PM c rPGM (W ) ∼ = Cc∞ (U \G) ⊗H(G) W W ∈ Rep(G). For V ∈ RepsM (M ) we may just as well take the bimodule Cc∞ (G/U )H(M )sM , and to get (73) the bimodule H(M )sM Cc∞ (U \G) is suitable. But if we want to obtain modules over H(M ⋊ R♯s )s , it is better to use the bimodules M Cc∞ (G/U )s := C ∞ (G/U )H(M )w(sM ) = Cc∞ (G/U )H(M ⋊ R♯s )s , w∈R♯s c M w(sM ) ∞ Cc∞ (U \G)s := Cc (U \G) = H(M ⋊ R♯s )s Cc∞ (U \G). ♯ H(M ) w∈Rs Indeed, we can rewrite (72) as IPGM (V ) ∼ = Cc∞ (G/U )H(M )sM ⊗H(M )sM V = Cc∞ (G/U )s ⊗H(M )sM V ∼ = Cc∞ (G/U )s ⊗H(M ⋊R♯ )s H(M ⋊ R♯s )s ⊗H(M )sM V s ♯ s H(M ⋊R ) ∼ = Cc∞ (G/U )s ⊗H(M ⋊R♯s )s IndH(M )sMs (V ). Similarly (73) translates to ♯ ⋊Rs IndM ◦ prsM ◦ rPGM = H(M ⋊ R♯s )s ⊗H(M )sM H(M )sM Cc∞ (U \G) ⊗H(G)s W M = H(M ⋊ R♯s )s Cc∞ (U \G) ⊗H(G)s W = Cc∞ (U \G)s ⊗H(G)s W. HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 27 These calculations entail that the bimodules Cc∞ (G/U )s and Cc∞ (U \G)s implement H(G)s ∼M H(M ⋊ R♯s )s . (76) These bimodules are naturally endowed with an action of X G (s), by pointwise multiplication of functions G → C. This action is obviously compatible with the group actions on H(G)s and H(M ⋊ R♯s )s , in the sense that γ · (f1 f2 ) = (γ · f1 )(γ · f2 ) and γ · (f2 f3 ) = (γ · f2 )(γ · f3 ) ♯ s s for γ ∈ X G (s), f1 ∈ H(G)s , f2 ∈ Cc∞ (G/U T ) , f3 ∈ H(M ⋊ Rs ) . Hence we may restrict (76) to functions supported on γ∈X G (s) ker γ, and we obtain (77) (Cc∞ (G/U )s )X G (s) G (Cc∞ (U \G)s )X (s) ⊗(H(M ⋊R♯ )s )X G (s) (Cc∞ (U \G)s )X G (s) s ⊗(H(G)s )X G (s) (Cc∞ (G/U )s )X G (s) G ∼ = (H(G)s )X (s) , G ∼ = (H(M ⋊ R♯s )s )X (s) . (b) Consider the idempotent p = |X L (s)|−1 X γ∈X L (s) γ ∈ C[X L (s)]. It is easy to see that the map (H(M )sM )X L (s) → p(H(M )sM ⋊ X L (s))p : a 7→ pap L is an isomorphism of algebras [Sol1, Lemma A.2]. Therefore (H(M )sM )X (s) is Morita equivalent with (H(M )sM ⋊ X L (s))p(H(M )sM ⋊ X L (s)), via the bimodules p(H(M )sM ⋊ X L (s)) and (H(M )sM ⋊ X L (s))p. Suppose that (H(M )sM ⋊ X L (s))p(H(M )sM ⋊ X L (s)) ( H(M )sM ⋊ X L (s). Then the quotient algebra  H(M )sM ⋊ X L (s) (H(M )sM ⋊ X L (s))p(H(M )sM ⋊ X L (s)) is nonzero. This algebra is a direct limit of unital algebras, so it has an irreducible module V on which it does not act as zero. We can regard V as an irreducible H(M )sM ⋊ X L (s)-module with pV = 0. For any π ∈ IrrsM (M ) we have X M (π) ⊂ X L (ω) since W (M, L) = Ws and by Lemma 2.4.d. By (16) and (18) the decomposition of Vπ over M ♯ Z(G) is governed by C[X M (π), κω ]. Now Clifford theory (see for example [Sol2, Appendix A]) says that, for any ρ ∈ Irr(C[X M (π), κω ]), H(M )sM ⋊X L (s) IndH(M )sM ⋊X M (π) (Vπ ⊗ ρ∨ ) is an irreducible module over H(M )sM ⋊X L (s). Moreover every irreducible H(M )sM ⋊ X L (s)-module is of this form, so we may take it as V . But by (16) (78) ρ appears in Vπ . Hence Vπ ⊗ ρ∨ has nonzero X L (ω)-invariant vectors and pV 6= 0. This contradiction shows that (79) (H(M )sM ⋊ X L (s))p(H(M )sM ⋊ X L (s)) = H(M )sM ⋊ X L (s). Recall from Lemma 3.6 that G X L (s) (H(M sM ⋊ R♯s )s )X (s) ∼ ⋊ R♯s = p(H(M )sM ⋊ Stab(s, P ∩ M ))p. = (H(M )sM ) 28 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD The bimodules p(H(M )sM ⋊ Stab(s, P ∩ M )) and (H(M )sM ⋊ Stab(s, P ∩ M ))p make it Morita equivalent with (H(M )sM ⋊ Stab(s, P ∩ M ))p(H(M )sM ⋊ Stab(s, P ∩ M )), which by (79) equals H(M )sM ⋊ Stab(s, P ∩ M ). (c) This follows from (75) and Lemma 3.4, upon applying ⋊Stab(s, P ∩ M ) everywhere. (d) First we want to show that (80) (esM H(M )esM )X L (s) ∼M (esM H(M )esM ) ⋊ X L (s). To this end we use the same argument as in part (b), only with esM H(M )esM instead of H(M )sM . Everything goes fine until (78). The corresponding statement in the present setting would be that every irreducible module of C[X M (π), κπ ] appears in esM Vπ . By 16 this is equivalent to saying that esM Vπ intersects every M ♯ -isotypical component of Vπ nontrivially, which is exactly Lemma 3.3.d. Therefore this version of (78) does hold. The analogue of (79) is now valid, and establishes (80). The bimodules for this Morita equivalence are p(esM H(M )esM ⋊ X L (s)) and (esM H(M )esM ⋊ X L (s))p. The same argument as after (79) makes clear how this implies the required Morita equivalence esM H(M )esM ⋊ Stab(s, P ∩ M ) ∼M (esM H(M )esM )X L (s) ⋊ R♯s .  From the above proof one can extract bimodules for the Morita equivalence (81) (esM H(M )esM )X L (s) ⋊ R♯s ∼M (H(M )sM )X L (s) ⋊ R♯s , namely (82) (H(M )sM esM )X L (s) ⋊ R♯s and (esM H(M )sM )X L (s) ⋊ R♯s . It seems complicated to prove directly that these are Morita bimodules, without the detour via parts (b) and (c) of Proposition 3.8. 3.3. Passage to the derived group. We study how Hecke algebras for G♯ and for G♯ Z(G) can be replaced by Morita equivalent algebras built from H(G). In the last results of this subsection we will show that a Morita equivalent subalgebra H(G♯ )s is isomorphic to subalgebras of H(G)s and of H(M ⋊ R♯s )s . Lemma 3.9. The algebra H(G♯ Z(G))s is Morita equivalent with (H(G)s )X G (s) . Proof. Let Cl be the l-th congruence subgroup of GLm (oD ), and put Cl′ = Cl ∩ G♯ Z(G). The group G♯ Z(G)Cl is of finite index in G, because Nrd(G♯ Z(G)Cl ) contains both F ×md and an open neighborhood Nrd(Cl ) of 1 ∈ F × . By Lemma 2.4.c we can choose l so large, that every element of X G (s) is trivial on Cl and that all representations in Reps (G) have nonzero Cl -invariant vectors. Let eCl ∈ H(Cl ) be the central idempotent associated to the trivial representation of Cl . It is known from [BeDe, §3] that (Cl ,triv) is a type, so the algebra H(G, Cl )s = eCl H(G)s eCl HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 29 of Cl -biinvariant functions in H(G)s is Morita equivalent with H(G)s . The Morita bimodules are eCl H(G)s and H(G)s eCl . Since X G (s) fixes eCl , these bimodules carry an X G (s)-action, which clearly is compatible with the actions on H(G)s and H(G, Cl )s . We can restrict the equations which T make them Morita bimodules to the subspaces of functions G → C supported on γ∈X G (s) ker γ. We find that the bimodules (eCl H(G)s )X tween G (s) and (H(G)s eCl )X (H(G)s )X (83) G (s) and G (s) provide a Morita equivalence be- (H(G, Cl )s )X G (s) . We saw in (42) that Irrs (G♯ Z(G)) is a union of Bernstein components, in fact a finite union by Lemma 2.2. Hence we may assume that every representation in Irrs (G♯ Z(G)) contains nonzero Cl′ -invariant vectors. As (Cl′ ,triv) is a type, that suffices for a Morita equivalence between H(G♯ Z(G))s (84) and H(G♯ Z(G), Cl′ )s . We may assume that the Haar measures on G and on G♯ Z(G) are chosen such that Cl and Cl′ get the same volume. Then the natural injection Cl′ \ G♯ Z(G)/Cl′ → Cl \ G/Cl provides an injective algebra homomorphism H(G♯ Z(G), Cl′ ) → H(G, Cl ), (85) whose image consists of the Irr(G/G♯ Z(G)Cl )-invariants in H(G, Cl ). Let B(G)l be the set of inertial equivalence classes for G corresponding to the category of Grepresentations that are generated by their Cl -invariant vectors. The finite group Irr(G/G♯ Z(G)Cl ) acts on it, and we denote the set of orbits by B(G)l / ∼. Now we can write M H(G♯ Z(G), Cl′ )s = H(G♯ Z(G), Cl′ ) ∼ = s∈B(G)l /∼ M G ♯ (H(G, Cl )s )X (s) . H(G, Cl )Irr(G/G Z(G)Cl ) = s∈B(G)l /∼ By considering the factors corresponding to one s on both sides we obtain an isomorphism G H(G♯ Z(G), Cl′ )s ∼ = (H(G, Cl )s )X (s) . To conclude, we combine this with (83) and (84).  The Morita equivalences in parts (a) and (d) of Proposition 3.8, for algebras associated to G♯ Z(G), have analogues for G♯ . For parts (b) and (c), which involve crossed products by Stab(s, P ∩ M ), this is not clear. Lemma 3.10. The algebra H(G♯ )s is Morita equivalent with (H(G)s )X G (s)X nr (G) and with (H(M )s )X L (s)X nr (G) ⋊ R♯s . Proof. By (40) we have (86) X (Z(G)) H(G♯ )s ∼ . = (H(G♯ Z(G))s ) nr As Xnr (G/Z(G)) ⊂ X L (s) ⊂ X G (s), every χ ∈ Xnr (Z(G)) extends in a unique way G to a character of H(G)X (s) . In other words, we can identify (87) Xnr (Z(G)) = Xnr (G) in Irr(G/G♯ )/X G (s). 30 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD All the bimodules involved in (83) and (84) carry a compatible action of (87). We can restrict T the proofs of these Morita equivalences to smooth functions supported on G1 = χ∈Xnr (G) ker χ. That leads to a Morita equivalence (H(G♯ Z(G))s )Xnr (Z(G)) ∼M (H(G)s )X G (s)X nr (G) . Let us take another look at the Morita equivalence (76), between H(G)s and H(M ⋊ R♯s )s . The argument between (76) and (77) also works with X G (s)Xnr (G) instead of X G (s), and provides a Morita equivalence (88) (H(G)s )X G (s)X nr (G) ∼M (H(M ⋊ R♯s )s )X G (s)X nr (G) . The isomorphism in Lemma 3.6 is Xnr (G)-equivariant, so it restricts to (H(M ⋊ R♯s )s )X G (s)X nr (G) L ∼ = (H(M )sM )X (s)Xnr (G) ⋊ R♯s .  We would like to formulate a version Lemma 3.10 with idempotents in H(G) and H(M ). Consider the types (KG , λG ⊗ γ) for γ ∈ X G (s). Lemma 3.11. Let γ, γ ′ ∈ X G (s). (a) The KG -representations λG ⊗ γ and λG ⊗ γ ′ are equivalent if and only if γ −1 γ ′ ∈ X L (s, λ). (b) For any a, a′ ∈ M the idempotents aeλG ⊗γ a−1 and a′ eλG ⊗γ ′ (a′ )−1 are orthogonal if γ −1 γ ′ ∈ X G (s) \ X L (s). Proof. (a) Suppose first that γ −1 γ ′ ∈ X G (s)\X L (s). Then (K, λ) and (K, λ⊗ γ) are types for different Bernstein components of M , so λ and λ ⊗ γ are not equivalent. As both λG and γ are trivial on KG ∩ U and on KG ∩ U , this implies that λG and λG ⊗ γ are not equivalent either. Now suppose that γ ∈ X L (s). By the definition of Stab(s, λ), the K-representations λ and λ ⊗ γ are equivalent if and only if γ ∈ Stab(s, λ). By the same argument as above, this statement can be lifted to λG and λG ⊗ γ. (b) Consider the idempotents aeλ⊗γ a−1 and a′ eλ⊗γ ′ (a′ )−1 in H(M ). They belong ′ to the subalgebras H(M )sM ⊗γ and H(M )sM ⊗γ , respectively. Since X L (s) = X M (s) and γX L (s) 6= γ ′ X L (s), these are two orthogonal ideals of H(L). In particular the two above idempotents are orthogonal. Let hKG ∩ U i denote the idempotent, in the multiplier algebra of H(G), which corresponds to averaging over the group KG ∩ U . Then aeλG ⊗γ a−1 = ahKG ∩ U ihKG ∩ U ieλ⊗γ a−1 = ha(KG ∩ U )a−1 iha(KG ∩ U )a−1 iaeλ⊗γ a−1 Similarly a′ eλG ⊗γ (a′ )−1 = aeλ⊗γ ′ a−1 ha(KG ∩ U )a−1 iha(KG ∩ U )a−1 i Now we see from the earlier orthogonality result that aeλG ⊗γ a−1 a′ eλG ⊗γ ′ (a′ )−1 = 0. Generalizing (59) we define X G (s/λ) = X G (s)/(X L (s) ∩ Stab(s, λ)).  HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 31 By Lemma 3.11 the elements X eµG := eλG ⊗γ ∈ H(G), γ∈X G (s/λ) X e♯λG := aeµG a−1 ∈ H(G), (89) a∈[L/Hλ ] X X e♯λ := aeλ⊗γ a−1 ∈ H(M ) G a∈[L/Hλ ] γ∈X (s/λ) are idempotent. We will show that the latter two idempotents see precisely the categories of representations of G and G♯ (resp. M and M ♯ ) associated to s. However, in general they do not come from a type, for the elements a ∈ [L/Hλ ] and cγ ∈ L need not lie in a compact subgroup of G. Lemma 3.12. Let (π, Vπ ) ∈ Irrs (G). Then e♯λG Vπ intersects every G♯ -isotypical component of Vπ nontrivially. Proof. The twisted group algebra C[X G (π), κπ ] acts on Vπ via intertwining operators. In view of (16), we have to show that e♯λG Vπ intersects the ρ-isotypical part of Vπ nontrivially, for every ρ ∈ Irr(C[X G (π), κπ ]). Choose χ ∈ Xnr (L) such that π is a subquotient of IPG (ω ⊗ χ). Then X G (π) ∩ X L (s, λ) ⊂ X L (ω ⊗ χ). As observed in the proof of Lemma 3.3.d, every irreducible representation of C[X G (π) ∩ X L (s, λ), κω⊗χ ] appears in e♯λG Vπ . The idempotents {aeλG ⊗γ a−1 : a ∈ [L/Hλ ], γ ∈ X G (s)} (90) are invariant under X L (s, λ) because eλG is, and they are mutually orthogonal by Lemma 3.11. As these idempotents sum to e♯λG , it follows that every every irreducible representation of C[X G (π) ∩ X L (s, λ), κω⊗χ ] already appears in one subspace aeλG ⊗γ a−1 Vπ . The quotient group X G (π)/X G (π) ∩ X L (s, λ) permutes the set of idempotents (90) faithfully. With Frobenius reciprocity we conclude that every irreducible representation of C[X G (π), κπ ] appears in e♯λG Vπ .  Lemma 3.13. (a) The algebras e♯λG H(G)X G (H(G)s )X (s) (b) G (s) e♯λG = (e♯λG H(G)e♯λG )X G (s) and are Morita equivalent. G ∼M (e♯λG H(G)e♯λG )X (s)Xnr (G) . G (H(G)s )X (s)Xnr (G) Proof. (a) Because all the types (KG , λG ⊗γ) are for the same Bernstein component s, the idempotent e♯λG sees precisely the category of representations Reps (G). Therefore the bimodules e♯λG H(G) and H(G)e♯λG implement a Morita equivalence H(G)s ∼M e♯λG H(G)e♯λG . (91) The same reasoning as in parts (b) and (c) of Proposition 3.8 establishes Morita equivalences (H(G)s )X G (s) ∼M H(G)s ⋊ X G (s) ∼M (e♯λG H(G)e♯λG ) ⋊ X G (s). To get from the right hand side to (e♯λG H(G)e♯λG )X sition 3.8.d. This is justified by Lemma 3.12. G (s) we follow the proof of Propo- 32 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD (b) The above argument also shows that the Morita equivalence of part (a) is implemented by the bimodules e♯λG H(G)X (92) G (s) H(G)X and G (s) e♯λG . These bimodules are endowed with actions of Xnr (G). Taking invariants under these group actions amounts to considering only functions supported on G1 . We note that e♯λG is supported on G1 and that this is a normal subgroup of G. Therefore the equations that make (92) Morita bimodules restrict to analogous equations for functions supported on G1 , which provides the desired Morita equivalence.  Proposition 3.14. There are algebra isomorphisms ♯ ♯ ♯ (a) e♯λG H(G)e♯λG ∼ = eλ H(M ⋊ Rs )eλ , L G G ♯ ♯ ♯ ♯ (b) (e♯λG H(G)e♯λG )X (s) ∼ = (esM H(M )esM )X (s) ⋊ Rs , = (eλ H(M ⋊ Rs )eλ )X (s) ∼ (c) between the three algebras of Xnr (G)-invariants in (b). Moreover the isomorphisms in (b) and (c) can be chosen such that, for every a1 , a2 ∈ [L/Hλ ], they restrict to linear bijections X (a1 eµG H(G)eµG a−1 2 ) G (s) X (a1 eµG H(G)eµG a−1 2 ) G (s)X X ←→ (a1 eµ H(M )eµ a−1 2 ) nr (G) L (s) ⋊ R♯s , X ←→ (a1 eµ H(M )eµ a−1 2 ) L (s)X nr (G) ⋊ R♯s . Proof. For any γ ∈ X L (s) and w ∈ R♯s there exists a γ ′ ∈ X G (s) such that w(λ⊗γ) ∼ = λ ⊗ γ ′ as representations of K. Hence (93) e♯λ H(M ⋊ R♯s )e♯λ ∩ H(M )w(sM ) = wesM w−1 H(M )wesM w−1 , M
s s −1 e♯λ H(M ⋊ R♯s )e♯λ = ⋊ R♯s . ♯ weM H(M )eM w w∈Rs We note that the right hand side of (93) is isomorphic to esM H(M )esM ⊗ EndC (CR♯s ) (94) The equality (93) also shows that M G (e♯λ H(M ⋊ R♯s )e♯λ )X (s) = w∈R♯s (wesM H(M )esM w−1 )X L (s) ⋊ R♯s
X G (s)/X L (s) . We can apply the same argument from the proof of Lemma 3.6 to the right hand side, which gives a canonical isomorphism G L (95) (e♯λ H(M ⋊ R♯s )e♯λ )X (s) ∼ = (esM H(M )esM )X (s) ⋊ R♯s . P Notice that for a ∈ L/Hλ the idempotents aeµ a−1 and a γ∈X G (s/λ) eλ⊗γ a−1 are invariant under Stab(s, P ∩ M ) and X G (s), respectively. Hence we can write (96) M G G (e♯λG H(G)e♯λG )X (s) = (a1 eµG H(G)eµG a−1 )X (s) , 2 a1 ,a2 ∈[L/Hλ ] M s s X L (s) ♯ X L (s)Xnr (G) (eM H(M )eM ) ⋊ Rs = (a1 eµ H(M )eµ a−1 ⋊ R♯s . 2 ) a1 ,a2 ∈[L/Hλ ] It is clear from the proof of Lemma 3.6 that the isomorphism (95) respects these decompositions. Moreover (95) is equivariant with respect to the actions of Xnr (G), so it restricts to G L (e♯ H(M ⋊ R♯ )e♯ )X (s)Xnr (G) ∼ = (es H(M )es )X (s)Xnr (G ⋊ R♯ . λ s λ M M We have proved the second isomorphism of part (b) and of part (c). s HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 33 For every γ ∈ X G (s/λ) and a ∈ [L/Hλ ] one has −1 ∼ aeλG ⊗γ a−1 H(G)aeλG ⊗γ a−1 = aα−1 = eλG ⊗γ H(G)eλG ⊗γ . γ (eλG ⊗γ H(G)eλG ⊗γ )a By Lemma 3.11 these are mutually orthogonal subalgebras of e♯λG H(G)e♯λG . The inclusion aeλG ⊗γ a−1 H(G)aeλG ⊗γ a−1 → e♯λG H(G)e♯λG is a Morita equivalence and for all V ∈ Rep(G)s : M M e♯λG V = aeλG ⊗γ a−1 V. a∈[L/Hλ ] γ∈X G (s/λ) It follows that the aeλG ⊗γ a−1 form the idempotent matrix units in some subalgebra Mn (C) ⊂ e♯λG H(G)e♯λG , and that e♯λG H(G)e♯λG ∼ = eλG H(G)eλG ⊗ Mn (C) where n = |L/Hλ | |X G (s/λ)|. The same argument shows that esM H(M )esM ∼ = eλ H(M )eλ ⊗ Mn′ (C), where n′ = |L/Hλ | |X L (s/λ)|. Since (KG , λG ) is a cover of (K, λ), eλ H(M )eλ ∼ = eλ H(G)eλ . G G By Lemma 3.11 n′ |R♯s | = |L/Hλ | |X L (s/λ)| |R♯s | = |L/Hλ | |X G (s/λ)| = n. With (94) we deduce that (97) ♯ ♯ e♯λ H(M ⋊ R♯s )e♯λ ∼ = eλG H(G)eλG ⊗ Mn (C) ∼ = eλG H(G)eλG , proving part (a). It entails from (76) that (98) e♯λG Cc∞ (G/U )e♯λ and e♯λ Cc∞ (U \G)e♯λG are bimodules for a Morita equivalence (99) e♯λ H(M ⋊ R♯s )e♯λ ∼M e♯λG H(G)e♯λG . But by (97) these algebras are isomorphic, so the bimodules are free of rank 1 over both algebras. Similarly, it follows from (77) that (100) (Cc∞ (G/U )s )X G (s) and (Cc∞ (U \G)s )X G (s) are bimodules for a Morita equivalence between (H(G)s )X (s) and (H(M ⋊R♯s )s )X (s) . By Lemma 3.13, Proposition 3.8 and (95) there is a chain of Morita equivalences G (101) (e♯λG H(G)e♯λG )X G (s) ∼M (H(G)s )X G (s) G ∼M (H(M ⋊ R♯s )s )X ∼M (e♯λ H(M ⋊ R♯s )e♯λ )X G (s) G (s) . The respective Morita bimodules are given by (92), (100) and (82). In relation to (95) we can rewrite (82) as (102) e♯λ (H(M ⋊ R♯s )s )X G (s) and (H(M ⋊ R♯s )s )X G (s) e♯λ . 34 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD It follows that Morita bimodules for the composition of (101) are (e♯λG Cc∞ (G/U )e♯λ )X (103) G (s) and (e♯λ Cc∞ (U \G)e♯λG )X G (s) . As the modules (100) are free of rank 1 over both the algebras (99), and the actions of X G (s) on (103) and the involved algebras come from the action on functions G → C, the modules (103) are again free of rank 1 over (e♯λG H(G)e♯λG )X (104) G (s) and (e♯λ H(M ⋊ R♯s )e♯λ )X G (s) . Therefore these two algebras are isomorphic. Since the idempotents aeµG a−1 and P G a γ∈X G (s/λ) eλ⊗γ a−1 are X G (s)-invariant, e♯λG H(G)e♯λG )X (s) and the bimodules (103) can be decomposed in the same way as (96). It follows that the isomorphism between the algebras in (104), as just constructed from (103), respects the decompositions indexed by a1 , a2 ∈ [L/Hλ ]. This settles part (b). It remains to prove the first isomorphism of part (c), but here we encounter the problem that the isomorphism between the algebras (104) is not explicit. In particular we do not know for sure that it is equivariant with respect to Xnr (G). Nevertheless, we claim that the chain of Morita equivalences (101) remains valid upon taking Xnr (G)-invariants. For the first equivalence that is Lemma 3.13.b and for the second equivalence it was checked in (88). For the third equivalence we can use the same argument as for the first, the equations making (102) Morita bimodules can be restricted to functions G → C supported on G1 G♯ . Composing these three steps, we obtain (e♯λG H(G)e♯λG )X (105) G (s)X nr (G) ∼M (e♯λ H(M ⋊ R♯s )e♯λ )X G (s)X nr (G) . with Morita bimodules (e♯λG Cc∞ (G/U )e♯λ )X (106) G (s)X nr (G) and (e♯λ Cc∞ (U \G)e♯λG )X G (s)X nr (G) . Since the modules (103) are free of rank one over the algebras (104), the modules (106) are free of rank one over both the algebras in (105). Therefore these two algebras are isomorphic. The isomorphism respects the decompositions indexed by a1 , a2 ∈ [L/Hλ ] for the same reasons as in part (b).  We normalize the Haar measures on G, G♯ and G♯ Z(G) such that KG and KG ∩G♯ and KG ∩G♯ Z(G) have the same volume. Consider e♯λG ∈ H(G) as a function G → C and let e♯λ G♯ (resp. e♯λ G♯ Z(G) e♯λ (107) ) be its restriction to G♯ (resp. G♯ Z(G)). We have G♯ Z(G) ∈ H(G♯ Z(G)) and e♯λ G♯ Theorem 3.15. The element e♯λ G♯ Z(G) e♯λ G♯ Z(G) H(G♯ Z(G))e♯λ G♯ Z(G) ∈ H(G♯ ). ∈ H(G♯ Z(G)) is idempotent and G L ♯ ♯ ∼ = eλG H(G)X (s) eλG ∼ = (esM H(M )esM )X (s) ⋊ R♯s . These algebras are Morita equivalent with H(G♯ Z(G))s and with (H(G)s )X G (s) . Proof. Consider the l-th congruence Cl ⊂ GLm (oD ), as in the proof of Lemma 3.9. We choose the level l so high that all representations in Reps (G) have nonzero Cl fixed vectors and that e♯λG is Cl -biinvariant. Put Cl′ = Cl ∩ G♯ Z(G). The proof of HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 35 Lemma 3.9 shows that the algebra isomorphism H(G♯ Z(G), Cl′ )s → (H(G, Cl )s )X (108) coming from (85) maps e♯λ G (s) to e♯λG . As e♯λG is idempotent, so is e♯λ G♯ Z(G) G♯ Z(G) . It follows that (108) restricts to an isomorphism (109) e♯λ G♯ Z(G) H(G♯ Z(G))e♯λ = e♯λ G♯ Z(G) G♯ Z(G) H(G♯ Z(G), Cl′ )s e♯λ G♯ Z(G) G (s) e♯λG (H(G, Cl )s )X → e♯λG = e♯λG H(G)X G (s) e♯λG . The second isomorphism of the lemma is Proposition 3.14.b. By Lemma 3.13 these G algebras are Morita equivalent with (H(G)s )X (s) , and by Lemma 3.9 also with H(G♯ Z(G))s .  Theorem 3.16. The element e♯λ G♯ e♯λ ♯ H(G♯ )e♯λ G♯ G ∈ H(G♯ ) is idempotent and G L ♯ ♯ ∼ = eλG H(G)X (s)Xnr (G) eλG ∼ = (esM H(M )esM )X (s)Xnr (G) ⋊ R♯s . These algebras are Morita equivalent with H(G♯ )s and with (H(G)s )X G (s)X nr (G) . Proof. Recall that (KG , λG ) is a type for the single Bernstein component s. The representations in Reps (G) contain only one character of Z(G) ∩ G1 , so we must have Z(G) ∩ KG = Z(G) ∩ G1 = o× F · 1m . Because λG is irreducible as a representation of KG , Z(G) ∩ KG acts on it by a character, say ζλ . Endow Z(G) with the Haar measure for which Z(G)∩KG gets volume |Z(G)∩G♯ |. There is an equality e♯λ ♯ = e♯λ ♯ eζλ G Z(G) of distributions on G1 , ζλ ). Then (110) G♯ Z(G), e♯λ ♯ H(G♯ )e♯λ G G♯ G where eζλ denotes the idempotent associated to (Z(G)∩ → e♯λ G♯ Z(G) H(G♯ Z(G))e♯λ G♯ Z(G) : f 7→ f eζλ is an injective algebra homomorphism with image e♯λ G♯ Z(G) H(G♯ (Z(G) ∩ G1 ))e♯λ G♯ Z(G) This is precisely the subalgebra of e♯λ G♯ Z(G) H(G♯ Z(G))e♯λ . G♯ Z(G) which is invariant under Xnr (G♯ Z(G)). Under the isomorphism (109) it corresponds to e♯λG H(G)X G (s)X nr (G) e♯λG . That algebra is isomorphic to (esM H(M )esM )X L (s)X nr (G) ⋊ R♯s G by Proposition 3.14.c and Morita equivalent to (H(G)s )X (s)Xnr (G) by Lemma 3.13.b. In Lemma 3.10 we already observed that this last algebra is Morita equivalent with H(G♯ )s .  36 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD 4. The structure of the Hecke algebras 4.1. Hecke algebras for general linear groups. Via the map χ 7→ ω ⊗ χ we identify Ts with the complex torus Xnr (L)/Xnr (L, ω). This gives us the lattices X ∗ (Ts ) and X∗ (Ts ) of algebraic characters and cocharacters, respectively. We emphasize that this depends on the choice of the basepoint ω of Ts . Under the conditions 1.1, any other basepoint is the form ω ′ = ω ⊗ χ′ where χ′ ∈ Xnr (L)Ws . There is a natural isomorphism x 7→ x′ from X ∗ (Ts ) with respect to ω to X ∗ (Ts ) with respect to ω ′ . As functions on Ts , it works out to (111) x′ (ω ⊗ χ) = x(ω ⊗ χ)x(ω ⊗ χ′ )−1 . The inertial equivalence class s comes not only with the torus Ts and the group Ws , but also with a root system Rs ⊂ X ∗ (Ts ), whose Weyl group is Ws . From (111) we see that the character x is independent of the choice of a basepoint of Ts if it is invariant under Xnr (L)Ws , that is, if x lies in the lattice ZRs spanned by Rs . Let H(X ∗ (Ts ) ⋊ Ws , qs ) denote the affine Hecke algebra associated to the root datum (X ∗ (Ts ), X∗ (Ts ), Rs , Rs∨ ) and the parameter function qs as in [Séc]. It has a standard basis {[x] : x ∈ X ∗ (Ts ) ⋊ Ws }, with multiplication rules described first by Iwahori and Matsumoto [IwMa]. We remark that here qs is not a just one real number, but a collection of parameters qs,i > 0, one for each factor Mi of M , or equivalently one for each irreducible component of the root system Rs . The parameter qs has a natural extension to a map qs : X ∗ (Ts ) ⋊ Ws → R>0 , see [Lus, §1]. On the part of X ∗ (Ts ) that is positive with respect to P ∩ M it can be defined as follows. Since Ts is a quotient of Xnr (L), X ∗ (Ts ) is naturally isomorphic to a subgroup of L/L1 . In this way qs corresponds to δu−1 , the inverse of the modular character for the action of L on the unipotent radical of P ∩ M . Let us recall the Bernstein presentation of an affine Hecke algebra [Lus, §3]. For x ∈ X ∗ (Ts ) positive with respect to P ∩ M we write (112) θx := qs (x)−1/2 [x] = δu1/2 (x)[x]. The map x 7→ θx can be extended in a unique way to a group homomorphism X ∗ (Ts ) → H(X ∗ (Ts ) ⋊ Ws , qs )× [Lus, 2.6], for which we use the same notation. By [Lus, Proposition 3.7] {θx [w] : x ∈ X ∗ (Ts ), w ∈ Ws } is a basis of H(X ∗ (Ts ) ⋊ Ws , qs ). Furthermore the span of the θx is a subalgebra A isomorphic to C[X ∗ (Ts )] ∼ = O(Ts ) and the span of the [w] with w ∈ Ws is the Iwahori–Hecke algebra H(Ws , qs ). The multiplication map (113) A ⊗ H(Ws , qs ) → H(X ∗ (Ts ) ⋊ Ws , qs ) is a linear bijection. The commutation relations between these two subalgebras are known as the Bernstein–Lusztig–Zelevinsky relations. Let α ∈ Rs be a simple root, with corresponding reflection s ∈ Ws . By [Lus, Proposition 3.6], for any x ∈ X(Ts ) (114) θx [s] − [s]θs(x) = (qs (s) − 1)(θx − θs(x) )(1 − θ−α )−1 ∈ A. Since the elements [s] generate H(Ws , qs ), this determines the commutation relations for A with all [w] (w ∈ Ws ). It follows from (114) that (115) Z(H(X ∗ (Ts ) ⋊ Ws , qs )) = AWs . HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 37 In view of (111), (113) and (114), we can also regard H(X ∗ (Ts ) ⋊ Ws , qs ) as the algebra whose underlying vector space is (116) O(Ts ) ⊗ H(Ws , qs ) and whose multiplication satisfies (117) f [s] − [s](s · f ) = (qs (s) − 1)(f − (s · f ))(1 − θ−α )−1 f ∈ O(Ts ), with respect to the canonical action of Ws on O(Ts ). The advantage is that, written in this way, the multiplication does not depend on the choice of a basepoint ω ∈ Ts used to define X ∗ (Ts ). We will denote this interpretation of H(X ∗ (Ts ) ⋊ Ws , qs ) by H(Ts , Ws , qs ). Let ̟D be a uniformizer of D. Consider the group of diagonal matrices in L all whose diagonal entries are powers of ̟D and whose components in each Li are multiples of the identity. It can be identified with a sublattice of X ∗ (Xnr (L)). The lattice X ∗ (Ts ) can be represented in a unique way by such matrices, say by the group ∗ (T ) ⊂ L. X^ s The next result is largely due to Sécherre [Séc]. Theorem 4.1. For every (w, γ) ∈ Stab(s, P ∩ M ) there are isomorphisms ew(λ )⊗γ H(L)ew(λ )⊗γ ∼ = H(L, w(λL ) ⊗ γ) ⊗ EndC (Vw(λ )⊗γ ) L L ew(λ)⊗γ H(M )ew(λ)⊗γ L ∼ = O(Ts ) ⊗ EndC (Vw(λ)⊗γ ), ∼ H(M, w(λ) ⊗ γ) ⊗ EndC (Vw(λ)⊗γ ) = ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vw(λ)⊗γ ). The first isomorphism is canonical, the second depends only on the choice of the ∗ (T )K parabolic subgroup P . The support of these algebras is, respectively, KL X^ s L ^ ∗ and K X (Ts )Ws K. Proof. Since all the types (K, w(λ) ⊗ γ) have the same properties, it suffices to treat the case (w, γ) = (1, 1). The first and third isomorphisms are instances of (6). The support of the algebras was determined in [Séc, §4]. Sécherre also proved that the remaining isomorphisms exist, but some extra work is needed to make them canonical. The L-representations ω ⊗ χ with χ ∈ Xnr (L) paste to an algebra homomorphism (118) FL : eλL H(L)eλL → O(Xnr (L)) ⊗ EndC (eλL Vω ), which is injective because these are all irreducible representations in RepsL (L). By [Séc, Théorème 4.6] eλL H(L)eλL is isomorphic to O(Ts ) ⊗ EndC (VλL ). Hence (119) eλ Vω ∼ = Vλ = Vλ L L and (118) restricts to a canonical isomorphism (120) FL : eλL H(L)eλL → O(Ts ) ⊗ EndC (VλL ). Here O(Ts ) is the centre of the right hand side, so it corresponds to H(L, λL ). Consider the isomorphism (121) eλ H(M )eλ ∼ = H(X ∗ (Ts ) ⋊ Ws , qs ) ⊗ EndC (Vλ ). from [Séc, Théorème 4.6]. It comes from H(X ∗ (Ts ) ⋊ Ws , qs ) ∼ = H(M, λ). We define (122) fx,λ ∈ H(M, λ) as the image of [x] under (121). 38 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Because (K, λ) is a cover of (KL , λL ), we may use the results of [BuKu3, §7]. By [BuKu3, Corollary 7.2] there exists a unique injective algebra homomorphism (123) tP,λ : H(L, λL ) → H(M, λ) which is compatible with the Jacquet restriction functor rPM∩M . We note that in [BuKu3] unnormalized Jacquet restriction is used, whereas we prefer the normalized version. Therefore our tP,λ equals tδ1/2 in the notation of [BuKu3, §7], where δu u denotes the modular character for the action of L on the unipotent radical of P ∩ M . Consider the diagram H(M, λ) → H(X ∗ (Ts ) ⋊ Ws , qs ) ∼ = H(Ts , Ws , qs ) ↑ tP,λ ↑iP,λ (124) H(L, λL ) → O(Ts ) ∼ = C[X ∗ (Ts )], where the upper map is (121) and the lower map comes from (120). The horizontal maps are isomorphisms and tP,λ is injective. We want to define the right vertical map iP,λ so that the diagram commutes. The construction of the upper map in [Séc, §4] shows that it is canonical on the subalgebra of H(X ∗ (Ts )⋊Ws , qs ) generated by the elements [s] with s ∈ X ∗ (Ts )⋊Ws a simple affine reflection. This subalgebra has a basis {[x] : x ∈ ZRs ⋊ Ws }, where ZRs is the sublattice of X ∗ (Ts ) spanned by the root system Rs . In particular the image fx,λ ∈ H(M, λ) of [x] with x ∈ ZRs ⋊ Ws is defined canonically. By [Séc, Théorème 4.6] the remaining freedom for (121) boils down to, for each factor Mi of M , the choice of a nonzero element in a one-dimensional vector space. This is equivalent to the freedom in the choice of the basepoint ω of Ts . Take a x ∈ X ∗ (Ts ) which is positive with respect to P ∩ M , and let fx,λL be the corresponding element of H(L, λL ). For such elements tP,λ is described explicitly by [BuKu3, Theorem 7.2]. In our notation tP,λ (fx,λL ) = tδ1/2 (fx,λL ) = δu1/2 (x)fx,λ . u Suppose that furthermore x ∈ ZRs . From the compatibility of tP,λ with normalized Jacquet restriction one sees that, in order that the diagram commutes, it is necessary that (125) iP,λ (x) = θx . The condition (125) determines iP,λ (x) for all x ∈ ZRs . Now every way to extend iP,λ to the whole of C[X ∗ (Ts )] corresponds to precisely one choice of an isomorphism (121). Thus we can normalize (121) by requiring that (125) holds for all x ∈ X ∗ (Ts ) which are positive with respect to P ∩ M . In effect, we defined iP,λ to be the identity of O(Ts ) with respect to the isomorphisms A∼ = C[X ∗ (Ts )] ∼ = O(Ts ). So we turned (125) into an algebra homomorphism iP,λ : O(Ts ) → H(Ts , Ws , qs ). A priori it depends on the choice of a basepoint of Ts , but since we use the same basepoint on both sides and by (111), any other basepoint would produce the same map iP,λ . Thus (121) becomes canonical if we interpret the right hand side as H(Ts , Ws , qs ) ⊗ EndC (Vλ ).  HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 39 4.2. Projective normalizers. We will subject the algebra esM H(M )esM to a closer study, and describe its structure explicitly. At the same time we investigate how close eµ and esM are to the idempotent of a type. A natural candidate for such a type would involve the projective normalizer of (K, λ), but unfortunately it will turn out that this is in general not sufficiently sophisticated. Recall the groups defined in (58) and (59) and consider the vector spaces X X Vµ1 = Vµ1 := ew(λL )⊗γ Vω = eλL ⊗γ Vω , L (126) (w,γ)∈Stab(s,P ∩M )1 Vµ = VµL := X γ∈X L (s)1 X ew(λL )⊗γ Vω = eλL ⊗γ Vω . γ∈X L (s) (w,γ)∈Stab(s,P ∩M ) They carry in natural way representations of KL , namely M M µ1L = w(λL ) ⊗ γ = (w,γ)∈Stab(s,P ∩M )1 /Stab(s,λ) (127) M µL = λL ⊗ γ, γ∈X L (s/λ)1 w(λL ) ⊗ γ = M λL ⊗ γ. γ∈X L (s/λ) (w,γ)∈Stab(s,P ∩M )/Stab(s,λ) We lift them to representations M µ1 = λ ⊗ γ, γ∈X L (s/λ)1 M µ= λ⊗γ L (128) γ∈X (s/λ) of K by making it trivial on K ∩ N and on K ∩ N . In particular µ1L is the restriction of µ1 to K ∩ L. They relate to the idempotent esM by X X X aeγ⊗µ1 a−1 = aeµ a−1 = esM . a∈[L/Hλ ] γ∈X L (s/λ) a∈[L/Hλ ] It will turn out that eµ1 ∈ H(K) is the idempotent of a type, for a compact open subgroup of M that contains K. The normalizer of the pair (KL , λL ) is N (KL , λL ) := {m ∈ NL (KL ) | m · λL ∼ = λL }. ∗ (T )K = K X ∗ (T ). ^ Lemma 4.2. N (KL , λL ) = X^ s L L s Proof. By Theorem 4.1 (129) ∗ (T )K . N (KL , λL ) ⊂ KL X^ s L With conditions 1.1 and (53) we can be more precise. As discussed in the proof of Proposition 3.1, Y Y

∗ (T )) ei . ∗ (T )) ei = ∗ (T ) = KLi (D × 1Li ∩ X^ KLi (Li ∩ X^ (130) KL X^ s s s i i N ⊗ei As λL = i λLi , the group N (KL , λL ) can be factorized similarly. Consider any element of the form (131) ∗ (T ). ki zi with ki ∈ KLi , zi ∈ D × 1Li ∩ X^ s 40 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD The group KLi , called J(β, A) in [Séc], is made from a stratum in Li = GLmi (D), and therefore it is normalized by zi , see [Séc, §2.3]. Furthermore zi belongs to the support of eλi H(Li )eλi , so it normalizes (KLi , λi ). Knowing that, we can follow the proof of Proposition 3.1 with ki zi in the role of ci . It leads to the conclusion that ki zi ∈ N (KL , λL ). It follows that ∗ (T ) = X ^ ∗ (T )K ⊂ N (K , λ ). KL X^ s s L L L Combine this with (129).  Inspired by [BuKu2] we define two variants of the projective normalizer of (KL , λL ): P N (KL , λL ) := {m ∈ NL (KL ) | m · λL ∼ = λL ⊗ γ for some γ ∈ X L (s)}, P N 1 (KL , λL ) := P N (KL , λL ) ∩ L1 . Lemma 4.3. (a) (µ1L , Vµ1 ) extends to an irreducible representation of P N 1 (KL , µL ). L (b) (P N 1 (KL , µL ), µ1L ) is an [L, ω]L -type and [P N 1 (KL , λL ) : KL ] = [Stab(s, P ∩ M )1 : Stab(s, λ)] = |X L (s/λ)1 |. (c) (µL , VµL ) extends to an irreducible representation of P N (KL , λL ) and [P N (KL , λL ) : N (KL , λL )] = [Stab(s, P ∩ M ) : Stab(s, λ)] = |X L (s/λ)|. Proof. (a) Just as in (118), there is a canonical injective algebra homomorphism (132) FL : eµ1 H(L)eµ1 → O(Xnr (L)) ⊗ EndC (eµ1 Vω ). L L X L (s)1 For γ ∈ the element cγ ∈ 1 definition of L . Moreover L1 L from Proposition 3.1 maps to EndC (Vω ) by the eλL ⊗γ1 H(L)eλL ⊗γ2 = cγ1 eλL H(L)eλL c−1 γ2 , so by (120) the image of (132) is contained in O(Ts ) ⊗ EndC (Vµ1 ). L As the different idempotents eλL ⊗γ are orthogonal, M Vµ1 = eλL ⊗γ Vω , L γ∈X L (s/λ)1 M H(KL )eµ1 = eµ1 H(KL ) ∼ = L L L γ∈X (s/λ)1 EndC (eλL ⊗γ Vω ). Furthermore FL (C{cγ eµ1 : γ ∈ X L (s/λ)1 }) is a subspace of EndC (Vµ1 ) of dimension L L |X L (s/λ)1 |. So by the injectivity of (132) the algebra homomorphism (133) FL : H(KL )eµ1 ⊗ C{cγ eµ1 : γ ∈ X L (s/λ)1 } → EndC (Vµ1 ) L L L is bijective. Consider any m ∈ P N 1 (KL , λL ). It permutes the λL ⊗ γ with γ ∈ X L (s)1 , so it commutes with eµ1 . Also FL (meµ1 ) ∈ EndC (Vµ1 ) because m ∈ L1 . L L L So by the injectivity of (132) and the surjectivity of (133), meµ1 = f eµ1 for some L L 1 f ∈ H(KL · {cγ : γ ∈ X (s/λ) }). Consequently m ∈ KL · {cγ : γ ∈ X L (s/λ)1 } and (134) P N 1 (KL , λL ) = KL · {cγ : γ ∈ X L (s/λ)1 } L HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 41 Now (133) shows that Vµ1 = eµ1 Vω is an irreducible representation of (134). L L 1 (b) Since all the idempotents eλL ⊗γ are L-conjugate, the category RepµL (L) equals RepλL (L), and (P N 1 (KL , λL ), µ1L ) is a type for this factor of Rep(L). The claims about the indices follow from (134). (c) For every γ ∈ X L (s) the element cγ ∈ L is unique up to N (KL , λL ), so (135) P N (KL , λL ) = N (KL , λL ){cγ | γ ∈ X L (s)}. Together with Proposition 3.1 this proves the claims about [P N (KL , λL ) : N (KL , λL )]. Part (a), and the map (132) show that µ1L extends to an irreducible representation of P N 1 (KL , λ)N (KL , λL ). The same holds for γ ⊗ µ1L with γ ∈ X L (s). We have M VµL = Vγ⊗µ1 L γ∈X (s/λ) L as representations of P N 1 (KL , λL )N (KL , λL ), and these subspaces are permuted transitively by the cγ with γ ∈ X L (s). This and (135) show that VµL extends to an irreducible representation of P N (KL , λL ).  Lemma 4.3 has an analogue in M . To state it we rather start with the sets P N 1 (K, λ) := (K ∩ N )P N 1 (KL , λL )(K ∩ N ), P N (K, λ) := (K ∩ N )P N (KL , λL )(K ∩ N ). Lemma 4.4. (a) The multiplication map (K ∩ N ) × P N 1 (KL , λL ) × (K ∩ N ) → P N 1 (K, λ) is bijective and P N 1 (K, λ) is a compact open subgroup of M . (b) µ1 extends to an irreducible P N 1 (K, λ)-representation and (P N 1 (K, λ), µ1 ) is an sM -type. (c) P N (K, λ) is a group and the multiplication map (K ∩ N ) × P N (KL , λL ) × (K ∩ N ) → P N (K, λ) is bijective. Furthermore µ extends to an irreducible P N (K, λ)-representation. Remark. We will show later that P N 1 (K, λ) and P N (K, λ) are really the projective normalizers of (K, λ) in M 1 and M , respectively. Proof. (a) By [Séc, Proposition 5.3] the multiplication map (136) (K ∩ N ) × (K ∩ L) × (K ∩ N ) → K is a homeomorphism. By (134) and because cγ ∈ L1 normalizes K ∩ N and K ∩ N (see the proof of Proposition 3.1), the analogue of (136) for P N 1 (K, λ) holds as well. At the same time this shows that P N 1 (K, λ) is compact, for its three factors are. As P N 1 (KL , λL ) normalizes K ∩ N and K ∩ N this factorization also proves that P N 1 (K, λ) is a group. Since K is open in M , so is the larger group P N 1 (K, λ). (b) In view of Lemma 4.3.a and (128) we can extend µ1 to P N 1 (K, λ) by µ1 (nmn) := µ1L (m), where nmn is as in the decomposition from part (a). Then µ1 is irreducible because µ1L is. Because (K, w(λ) ⊗ γ) is an sM -type, for each (w, γ) ∈ Stab(s, P ∩ M ), the 1 category Repµ (M ) equals Repλ (M ) = RepsM (M ), and (P N 1 (K, λ), µ1 ) is an sM type. 42 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD (c) The first two claims can be shown in the same way as part (a), using (135) instead of (134). For the last assertion we employ Lemma 4.3.c and set µ(nmn) := µL (m), with respect to the factorization we just established.  Now we can determine the structure of esM H(M )esM and some related algebras. Theorem 4.5. (a) There exist canonical algebra isomorphisms e 1 H(L)e 1 ∼ = H(L, µ1 ) ⊗ EndC (V 1 ) ∼ = O(Ts ) ⊗ EndC (Vµ1 ). µL L µL µL ∗ (T )P N 1 (K , λ ) The support of the left hand side is P N 1 (KL , λL )X^ s L L (b) Part (a) extends to algebra isomorphisms eµ H(L)eµ ∼ = O(Ts ) ⊗ EndC (Vµ ), L L esL H(L)esL ∼ = O(Ts ) ⊗ EndC (Vµ ) ⊗ M|L/Hλ | (C). (c) There exist algebra isomorphisms ∼ H(M, µ1 ) ⊗ EndC (Vµ1 ) ∼ eµ1 H(M )eµ1 = = H(Ts , Ws , qs ) ⊗ EndC (Vµ1 ), which are canonical up to the choice of the parabolic subgroup P . The support ∗ (T )W P N 1 (K, λ). of the left hand side is P N 1 (K, λ)X^ s s (d) Part (c) extends to algebra isomorphisms eµ H(M )eµ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ), esM H(M )esM ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ) ⊗ M|L/Hλ | (C). Proof. (a) By the Morita equivalence of esL H(L)esL and eλL H(L)eλL , there are isomorphisms of esL H(L)esL -bimodules (137) esL H(L)esL ∼ = esL H(L)eλL ⊗eλL H(L)eλL eλL H(L)esL M M = a1 eλL ⊗γ1 a−1 1 H(L)eλL ⊗eλ L H(L)eλL eλL H(L)a2 eλL ⊗γ2 a−1 2 a1 ,a2 ∈[L/Hλ ] γ1 ,γ2 ∈X L (s/λ) = M M −1 −1 −1 a1 cγ1 eλ c−1 γ1 a1 H(L)eλL H(L)a2 cγ2 eλL cγ2 a2 M M −1 a1 cγ1 eλL H(L)eλL c−1 γ2 a2 . a1 ,a2 ∈[L/Hλ ] γ1 ,γ2 ∈X L (s/λ) = a1 ,a2 ∈[L/Hλ ] γ1 ,γ2 ∈X L (s/λ) Here the subalgebra eµ1 H(L)eµ1 corresponds to L L M c e H(L)eλL c−1 γ2 . L 1 γ1 λL γ1 ,γ2 ∈X (s/λ) In combination with (120) it follows that the canonical map (132) is an isomorphism (138) FL : eµ1 H(L)eµ1 → O(Ts ) ⊗ EndC (Vµ1 ). L L This and Theorem 4.1 imply that the support of eµ1 H(L)eµ1 is as indicated. NoL L tice that O(Ts ) is the commutant of EndC (Vµ1 ) in O(Ts ) ⊗ EndC (Vµ1 ). Hence it corresponds to H(L, µ1L ) under the canonical isomorphism e 1 H(L)e 1 ∼ = H(L, µ1 ) ⊗ EndC (Vµ1 ). µL µL L HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 43 (b) By (137) we can identify esL H(L)esL as a vector space with O M M a1 eµ1 ⊗γ1 H(L)eµ1 ⊗γ1 a−1 Ca3 γ3 1 . L L a1 ∈[L/Hλ ],γ1 ∈X L (s)/X L (s)1 a3 ∈[L/Hλ ],γ3 ∈X L (s)/X L (s)1 From (138) we get an isomorphism M (139) a1 eµ1 ⊗γ1 H(L)eµ1 ⊗γ1 a−1 1 → L L a1 ∈[L/Hλ ],γ1 ∈X L (s)/X L (s)1 M O(Ts ) ⊗ EndC (a1 Vµ1 ⊗γ1 ). L a1 ∈[L/Hλ ],γ1 ∈X L (s)/X L (s)1 Recall that VµL = eµL H(L)eλL ⊗eλ L H(L)eλL VλL = eµL H(L)eµ1 ⊗eµ1 H(L)eµ1 Vµ1 = L L L L M M cγ1 Vµ1 . cγ1 eµ1 ⊗ Vµ1 = L L γ1 ∈X L (s)/X L (s)1 γ1 L For γ3 ∈ X L (s) the choice of cγ3 is unique up N (KL , λL ), by Lemma 4.3.c. The particular shape (56) implies that it is in fact unique up to N (KL , λL )Ws , so (140) cγ3 cγ1 differs from cγ3 γ1 by an element of N (KL , λL )Ws . With Lemma 4.2 we deduce that left multiplication by cγ3 defines a bijection Vγ1 ⊗µ1 = cγ1 Vµ1 → cγ3 γ1 Vµ1 = Vγ3 γ1 ⊗µ1 L L L L which depends on ω ⊗ χ ∈ IrrsL (L) in an algebraic way. More precisely, (141) cγ3 eγ1 ⊗µ1 ∈ (O(Ts ))Ws ⊗ HomC (Vγ1 ⊗µ1 , Vγ3 γ1 ⊗µ1 ). L L L Consequently (139) extends to an algebra isomorphism (142) eµL H(L)eµL → O(Ts ) ⊗ EndC (VµL ). It is more difficult to see what (139) should look like for elements of [L/Hλ ]. For those we use a different, inexplicit argument. For each a ∈ [L/Hλ ] the inclusion aeµL a−1 H(L)aeµL a−1 → esL H(L)esL is a Morita equivalence, because the idempotents aeµL a−1 , esL and eλL all see exactly the same category of L-representations, namely RepsL (L). For every V ∈ RepsL (L) we have M esL V = aeµL a−1 V, a∈[L/Hλ ] where all the summands have the same dimension. It follows that ∼ eµ H(L)eµ ⊗ M|L/H | (C). es H(L)es = L L L L λ By (142) the right hand side is isomorphic to O(Ts ) ⊗ EndC (VµL ) ⊗ M|L/Hλ | (C) ∼ = O(Ts ) ⊗ EndC (VµL ). (c) Just like (137) there is an isomorphism of esM H(M )esM -bimodules M M −1 a1 cγ1 eλ H(M )eλ c−1 esM H(M )esM ∼ = γ2 a2 . a1 ,a2 ∈[L/Hλ ] γ1 ,γ2 ∈X L (s/λ) 44 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD For x ∈ X ∗ (Ts ) ⋊ Ws let fx,λ ∈ eλ H(M )eλ be the element that corresponds to [x] ∈ H(X ∗ (Ts ) ⋊ Ws , qs ) ∼ = H(Ts , Ws , qs ) via Theorem 4.1. The elements fx,λ commute with eλ H(K)eλ ∼ = EndC (Vλ ). It follows that the element (143) −1 −1 cγ fx,λ c−1 γ = cγ eλ fx,λ eλ cγ = eλ⊗γ cγ fx,λ cγ eλ⊗γ is independent of the choice of cγ in Proposition 3.1. As conjugation by cγ turns the commutative diagram (124) into the corresponding diagram for λ ⊗ γ, we have cγ fx,λ c−1 γ = fx,w(λ)⊗γ , (144) the image of [x] in ew(λ)⊗γ H(M )eλ⊗γ under the canonical isomorphisms from Theorem 4.1. For every x ∈ X ∗ (Ts ) ⋊ Ws we define X X (145) fx,µ1 := cγ fx,λ c−1 fx,λ⊗γ ∈ eµ1 H(M )eµ1 . γ = γ∈X L (s/λ)1 γ∈X L (s/λ)1 By (143) fx,µ1 commutes with eµ1 H(K)eµ1 and with the cγ for γ ∈ X L (s/λ)1 , so it commutes with eµ1 H(P N 1 (K, λ))eµ1 . By (137) and Theorem 4.1 M eµ1 H(M )eµ1 = Cfx,µ1 ⊗ eµ1 H(P N 1 (K, λ))eµ1 , ∗ x∈X (Ts )⋊Ws ∗ (T )W P N 1 (K, λ). and the support of this algebra is P N 1 (K, λ)X^ s s The orthogonality of the different idempotents eλ⊗γ implies that the fx,µ1 satisfy the same multiplication rules as the fx,λ . Hence the span of the fx,µ1 is a subalgebra of eµ1 H(M )eµ1 isomorphic with H(X ∗ (Ts ) ⋊ Ws , qs ). We constructed an algebra isomorphism (146) eµ1 H(M )eµ1 ∼ = H(X ∗ (Ts ) ⋊ Ws , qs ) ⊗ EndC (Vµ1 ). Since H(M, µ1 ) is the commutant of EndC (Vµ1 ) inside (147) H(M, µ1 ) ⊗ EndC (Vµ1 ) ∼ = eµ1 H(M )eµ1 , it corresponds to H(X ∗ (Ts ) ⋊ Ws ) under the isomorphisms (146) and (147). Tensored with the identity on EndC (Vλ ), tP,λ from (123) becomes a canonical injection (148) eλL H(L)eλL ∼ = H(L, λL ) ⊗ EndC (Vλ ) → H(M, λ) ⊗ EndC (Vλ ) ∼ = eλ H(M )eλ . Since tP,λ and the analogous map tP,µ1 for µ1 are uniquely defined by the same property, they agree in the sense that (149) tP,λ ⊗ id = tP,µ1 ⊗ id on eλL H(L)eλL ∼ = O(Ts ) ⊗ EndC (Vλ ). Consequently the isomorphisms (146) and (138) fit in a commutative diagram (150) H(M, µ1 ) → H(X ∗ (Ts ) ⋊ Ws , qs ) ∼ = H(Ts , Ws , qs ) ↑iP,µ1 ↑ tP,µ1 H(L, µ1L ) → O(Ts ) ∼ = C[X ∗ (Ts )], Here iP,µ1 is defined like iP,λ , see (125). In this sense (146) is canonical. (d) Part (c) works equally well with γ1 ⊗ µ1 instead of µ1 . For all γ1 ∈ X L (s) HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS together that gives a canonical isomorphism (151) M eγ1 ⊗µ1 H(M )eγ1 ⊗µ1 → H(Ts , Ws , qs ) ⊗ γ1 ∈X L (s)/X L (s)1 γ1 M 45 EndC (Vγ1 ⊗µ1 ). ∈X L (s)/X L (s)1 The formula (141) defines an element of H(Ts , Ws , qs ) ⊗ EndC (Vµ ) which commutes with H(Ts , Ws , qs ). Therefore we can extend (151) to isomorphisms eµ H(M )eµ → H(Ts , Ws , qs ) ⊗ EndC (Vµ ), esM H(M )esM → H(Ts , Ws , qs ) ⊗ EndC (Vµ ) in the same way as we did in the proof of part (b).  We note that a vector space basis of H(Ts , Ws , qs ) ⊂ eµ H(M )eµ is formed by the elements X X (152) fx,µ = cγ fx,λ c−1 = fx,λ⊗γ . γ γ∈X L (s/λ) γ∈X L (s/λ) We define the projective normalizer of (K, λ) in M as (153) {g ∈ NM (K) | g · λ ∼ = λ ⊗ γ for some γ ∈ X L (s)}. Using the explicit information gathered in the above proof, we can show that it is none other than P N (K, λ). Lemma 4.6. (a) (P N 1 (K, λ), µ1 ) is a cover of (P N 1 (KL , λL ), µ1L ). (b) P N (K, λ) equals the projective normalizer of (K, λ) in M . (c) P N 1 (K, λ) equals the projective normalizer of (K, λ) in M 1 . Proof. (a) For the definition of a cover we refer to [BuKu3, 8.1]. By Lemma 4.4 P N 1 (K, λ) ∩ L = P N 1 (KL , λL ) and by [Séc, Proposition 5.5] K admits an Iwahori decomposition with respect to any parabolic subgroup of M with Levi factor L. Hence P N 1 (K, λ) is also decomposed in this sense. The second condition for a cover says that µ|N (KL ,µL ) = µL , which is true by definition. The third condition is about the existence of an invertible “strongly positive” element in H(M, µ1 ). By [Séc, Proposition 5.5] H(M, λ) contains such an element, in the notation of the proof of Theorem 4.5 it corresponds to fx,λ for a suitable x ∈ X ∗ (Ts ). Then fx,µ1 and its image in H(M, µ1 ) have the correct properties. (b) By Lemma 4.4 P N (K, λ) is contained in this normalizer. Consider any g in the group (153). Its intertwining property entails that eµ geµ ∈ eµ H(M )eµ has inverse eµ g −1 eµ . From Theorem 4.1 we can see what the support of eµ H(M )eµ is, namely ∗ (T )W P N (K, λ). P N (K, λ)X^ s s Possibly adjusting g from the left and from the right by an element of P N (K, λ), we may assume that ∗ (T )W ⊂ N (L). g ∈ X^ s s M Then g also normalizes (KL , µL ). With Conditions 1.1 we see easily that every ∗ (T ), w ∈ W , we element of Ws normalizes (KL , µL ). Writing g = xw with x ∈ X^ s s find that x ∈ L normalizes (KL , µL ) as well. In other words x ∈ P N (KL , λL ) ⊂ P N (K, λ). 46 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD It follows that w ∈ Ws must also normalize (K, µ). By considering supports, one sees that there exist h1 , h2 ∈ EndC (Vµ ) such that eµ weµ and eµ w−1 eµ correspond to [w] ⊗ h1 and [w−1 ] ⊗ h2 respectively, under the isomorphism from Theorem 4.5. Moreover, because w lies in (153), eµ weµ · eµ w−1 eµ = eµ , [w] ⊗ h1 · [w−1 ] ⊗ h2 = [w] · [w−1 ] ⊗ h1 h2 = [1] ⊗ id. In particular [w][w−1 ] ∈ C[1]. The multiplication rules for H(Ws , qs ) [Séc, 2.5.3], applied with induction to the length of w ∈ Ws , show that this is only possible if ∗ (T ). w = 1. Consequently g = x ∈ P N (K, λ) ∩ X^ s (c) This follows immediately from (b).  Remark. In Lemma 4.6 we construct a sM -type with representation µ1 , but we do not succeed in finding a sM -type with representation µ. The obstruction appears to be that some of the representations λ ⊗ γ are conjugate in G, but not via an element of G1 . Examples 5.6 and 5.7 show that this can really happen when G is not split. Mainly for this reason we have been unable to construct types for all Bernstein components of G♯ . In the special case where all the KG -representations λG ⊗ γ with γ ∈ X G (s) are conjugate via elements of G1 , we can construct types for every Bernstein component t♯ ≺ s. We did not include this in the paper because it is quite some work and it is not clear how often these extra conditions are fulfilled. 4.3. Hecke algebras for the intermediate group. Recall from Proposition 3.8 and Lemma 3.9 that the algebras (esM H(M )esM )X L (s) ⋊ R♯s and (esM H(M )esM ) ⋊ Stab(s, P ∩ M ). are Morita equivalent with H(G♯ Z(G))s . In Theorem 3.15 we showed that the first one is even isomorphic to a subalgebra of H(G♯ Z(G))s determined by an idempotent. In Lemma 3.6 we saw that the actions of X L (s) and R♯s both come from the action α of Stab(s, P ∩ M ) defined in (71). Lemma 4.7. There is an equality M L (esM H(M )esM )X (s,λ) = a∈[L/Hλ ] (aeµ a−1 H(M )aeµ a−1 )X L (s,λ) , and this algebra is Stab(s, P ∩ M )-equivariantly isomorphic to M|X L (ω,Vµ )| L (eµ H(M )eµ )X (s,λ) . 1 L Remark. Here and below we use the notation n1 for the direct sum of n copies of something. Proof. As vector spaces esM H(M )esM = M a1 ,a2 ∈[L/Hλ ] a1 eµ H(M )eµ a−1 2 . On the other hand, for any H(M )sM -module V we have the decompositions M M (154) esM V = aeµ a−1 V = esM Vρ . a∈[L/Hλ ] ρ∈Irr(C[X L (s,λ)∩X L (ω),κω ]) HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 47 L s Here every aeµ a−1 V equals ρ∈Ia eM Vρ for a suitable collection Ia of ρ’s. If f ∈ esM H(M )esM is invariant under X L (s, λ) ∩ X L (ω), then it must stabilize each of the spaces esM Vρ , and therefore it preserve both the decompositions (154). In view of Theorem 4.5.d, this is only possible if M f∈ aeµ a−1 H(M )aeµ a−1 , a∈[L/Hλ ] which proves the desired equality. By Lemma 3.3.b conjugation by a ∈ [L/Hλ ] gives a Stab(s, P ∩ M )-equivariant isomorphism eµ H(M )eµ → aeµ H(M )eµ a−1 = aeµ a−1 H(M )aeµ a−1 . By Lemma 3.5 |L/Hλ | = |Irr(X L (ω, Vµ ))| and this equals |X L (ω, Vµ )| since we dealing with an abelian group.  It turns out that the direct sum decomposition from Lemma 4.7 can already be observed on the level of subalgebras of H(G): Lemma 4.8. There are algebra isomorphisms M|X L (ω,Vµ )| L L (eµ H(M )eµ )X (s) ⋊ R♯s ∼ = (esM H(M )esM )X (s) ⋊ R♯s 1 M G G ♯ ♯ ∼ (aeµG H(G)eµG a−1 )X (s) = (eλG H(G)eλG )X (s) = a∈[L/Hλ ] ∼ = M|X L (ω,Vµ )| 1 (eµG H(G)eµG )X G (s) . Proof. The first isomorphism is a direct consequence of Lemma 4.7 and the second is Proposition 3.14.b. As shown in Proposition 3.14, it can be decomposed as M M X G (s) X L (s) (a1 eµG H(G)eµG a−1 ) ←→ (a1 eµ H(M )eµ a−1 ⋊ R♯s . 2 2 ) a1 ,a2 ∈[L/Hλ ] a1 ,a2 ∈[L/Hλ ] But by Lemma 4.7 the summands with a1 6= a2 are 0 on the right hand side, so they are also 0 on the left hand side. This proves the equality in the lemma. The final isomorphism is given by (eµG H(G)eµG )X G (s) → (aeµG H(G)eµG a−1 )X G (s) : f 7→ af a−1 .  Recall from Theorem 3.15 that the middle algebra in Lemma 4.8 is isomorphic to e♯λ ♯ H(G♯ Z(G))e♯λ ♯ , which is Morita equivalent with H(G♯ Z(G))s . In G Z(G) G Z(G) the above direct sum decomposition also holds on this level. to formulate it, let eµG♯ Z(G) ∈ H(G♯ Z(G)) be the restriction of eµG : KG → C to G♯ Z(G) ∩ K. We define By our choice of Haar measures, eµG♯ Z(G) is idempotent. See also (107). Corollary 4.9. There are algebra isomorphisms M e♯λ ♯ H(G♯ Z(G))e♯λ ♯ = aeµG♯ Z(G) a−1 H(G♯ Z(G))aeµG♯ Z(G) a−1 G Z(G) a∈[L/Hλ ] G Z(G) ∼ = ∼ = M|X L (ω,Vµ )| 1 M|X L (ω,Vµ )| 1 eµG♯ Z(G) H(G♯ Z(G))eµG♯ Z(G) (H(Ts , Ws , qs ) ⊗ EndC (Vµ ))X L (s) ⋊ R♯s . 48 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Proof. The equality comes from Lemma 4.8 and Theorem 3.15. For all a ∈ [L/Hλ ] the map eµG♯ Z(G) H(G♯ Z(G))eµG♯ Z(G) → aeµG♯ Z(G) a−1 H(G♯ Z(G))aeµG♯ Z(G) a−1 : f 7→ af a−1 is an algebra isomorphism. The remaining isomorphism follows again from Lemma 4.8.  For every (w, γ) ∈ Stab(s, P ∩ M ) there exists a χγ ∈ Xnr (L) such that w(ω) ⊗ γ ∼ = ω ⊗ χγ ∈ Irr(L). (155) Here ω ⊗ χγ is unique, so χγ is unique up to Xnr (L, ω). We may and will assume that γ ∈ Xnr (L/L♯ Z(G)) = Xnr (L) ∩ X L (s). Q In view of the conditions 1.1, in particular L = i Lei i , we may simultaneously assume that Y ⊗e (157) χγ = χγ,i i . (156) χγ = γ if i Notice that with this choice χγ ∈ Xnr (L) is invariant under Ws = W (M, L). Via the map χ 7→ ω ⊗ χ, χγ determines a Ws -invariant element of Ts . Recall the bijection −1 −1 −1 J(γ, ω ⊗ χ−1 γ ) ∈ HomL (ω ⊗ χγ , w (ω) ⊗ γ ) from (44). It restricts to a bijection Vµ = eµ Vω⊗χ−1 → Vµ = eµ Vw−1 (ω)⊗γ −1 . γ Clearly (156) enables us to take (158) J(γ, ω ⊗ χ−1 γ ) = idVω if γ ∈ Xnr (L/L♯ Z(G)). Recall from (67) and (13) that (159) × J(γ, ω ⊗ χ−1 γ )|Vµ ∈ C idVµ if γ ∈ X L (ω, Vµ ). Lemma 4.10. The action α of Stab(s, P ∩ M ) on eµ H(M )eµ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ) preserves both tensor factors. On H(Ts , Ws , qs ) it is given by −1 α(w,γ) (θx [v]) = χ−1 γ (x)θw(x) [wvw ] x ∈ X ∗ (Ts ), v ∈ Ws , and on EndC (Vµ ) by −1 −1 α(w,γ) (h) = J(γ, ω ⊗ χ−1 γ ) ◦ h ◦ J(γ, ω ⊗ χγ ) . Furthermore X L (ω, Vµ ) is the subgroup of elements that act trivially. Remark. It is crucial that χγ is Ws -invariant and that w normalizes P ∩ M , for otherwise the above formulae would not define an algebra automorphism of H(Ts , Ws , qs ). This can be seen with the Bernstein presentation, in particular (114). HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 49 Proof. By definition, for all χ ∈ Xnr (L), f ∈ eµL H(L)eµL (160) (ω ⊗ χ)(α(w,γ) (f )) = (w−1 (ω ⊗ χγ −1 ))(f ) −1 −1 −1 −1 = J(γ, ω ⊗ χ−1 γ ) ◦ (ω ⊗ w (χ)χγ )(f ) ◦ J(γ, ω ⊗ χγ ) . For f with image in EndC (Vµ ) the unramified characters χ and w−1 (χ)χ−1 γ are of no consequence. Thus (160) implies the asserted formula for α(w,γ) on EndC (Vµ ). Since H(Ts , Ws , qs ) is the centralizer of EndC (Vµ ) in eµ H(M )eµ , it is also stabilized by Stab(s, P ∩ M ). By considering supports we see that (161) α(w,γ) ([x]) ∈ C[wxw−1 ] for all x ∈ X ∗ (Ts ) ⋊ Ws . For any simple reflection s ∈ Ws , wsw−1 is again a simple reflection in Ws , because w ∈ R♯s normalizes P ∩ M . In H(Ws , qs ) we have ([s] + 1)([s] − qs (s)) = 0 = ([wsw−1 ] + 1)([wsw−1 ] − qs (wsw−1 )), where qs (s), qs (wsw−1 ) ∈ R>1 . Since α(w,γ) is an algebra automorphism, we deduce that qs (s) = qs (wsw−1 ) and that α(w,γ) ([s]) = [wsw−1 ]. Every v ∈ Ws is a product of simple reflections si , and then [v] is a product of the [si ] in the same way. Hence α(w,γ) ([v]) = [wvw−1 ] for all v ∈ Ws . The formula (71) also defines an action α of Stab(s, P ∩ M ) on eµ H(L)eµ ∼ = O(Ts ) ⊗ EndC (Vµ ), L L which for similar reasons stabilizes EndC (Vµ ). Now we have two actions of Stab(s, P ∩ M ) on EndC (Vµ ), depending on whether we considere it as a subalgebra of eµL H(L)eµL or of eµ H(M )eµ . It is obvious from the definition of µ that these two actions agree. The maps (148) and (150) lead to a canonical injection iP,µ : O(Ts ) → H(Ts , Ws , qs ), L which is a restriction of the map γ∈X L (s)/X L (s)1 iP,µ1 ⊗γ . Since it is canonical, iP,µ commutes with the respective actions α(w,γ) . Now we take x ∈ X ∗ (Ts ) ⊂ O(Ts ) in (160). With a Morita equivalence we can replace ω ⊗ χ by the one-dimensional O(Ts )-representation [ω ⊗ χ] with character ω ⊗ χ ∈ Ts . Then (160) becomes −1 (162) [ω ⊗ χ](α(w,χ) (x)) = [ω ⊗ χ−1 γ w (χ)](x) −1 = [ω ⊗ w−1 (χ)](χ−1 γ (x)x) = [ω ⊗ χ](χγ (x)w(x)). ∗ Thus α(w,γ) (x) = χ−1 γ (x)w(x). For x ∈ X (Ts ) positive w(x) is also positive, as w normalizes P ∩ M . We obtain −1 α(w,γ) (θx ) = α(w,γ) (iP,µ (x)) = iP,µ (α(w,χ) (x)) = iP,µ (χ−1 γ (x)w(x)) = χγ (x)θw(x) . Since α(w,γ) is an algebra homomorphism, this implies the same formula for all x ∈ X ∗ (Ts ). It is clear from (159) and (160) that the group X L (ω, Vµ ) fixes every element of eµ H(M )eµ . Conversely, suppose that (w, γ) acts trivially. From the formulas for 50 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD the action on H(Ts , Ws , qs ) we see that w = 1 and χγ ∈ Xnr (L, ω), so γ ∈ X L (ω). Then we deduce from (160) that γ ∈ X L (ω, Vµ ).  Since we have a type with idempotent eµ1 but not with idempotent eµ , we would L L like to reduce (eµ H(M )eµ )X (s) to (eµ1 H(M )eµ1 )X (s) . The next lemma solves a part of the problem, namely the elements cγ that do not lie in L1 . Lemma 4.11. Let γ ∈ X L (s) be such that χ(cγ ) = 1 for all χ ∈ Xnr (L/L♯ Z(G)) ∩ Xnr (L, ω). ∗ (T ) such that x c ∈ L1 L♯ . Then there exists xγ ∈ X^ s γ γ Proof. Recall that every χ ∈ Xnr (L, ω) vanishes on Z(L). Hence Xnr (L/L♯ Z(G)) ∩ Xnr (L, ω) = Xnr (L/L♯ ) ∩ Xnr (L, ω), and cγ determines a character of Xnr (L/L♯ )/Xnr (L/L♯ ) ∩ Xnr (L, ω). (163) ∗ (T ) such that This is a subtorus of Ts ∼ = Xnr (L)/Xnr (L, ω), so we can find xγ ∈ X^ s −1 xγ restricts to the same character of (163) as cγ . Then \ xγ cγ ∈ ker χ = L1 L♯ .  ♯ χ∈Xnr (L/L ) In view of Lemma 4.2 we may replace cγ by xγ cγ as in Lemma 4.11. From now on we assume that this has been done for all γ to which Lemma 4.11 applies. Recall that were already assuming that cγ ∈ L1 whenever this is possible. This gives rise to groups X L (s)2 = {γ ∈ X L (s) | cγ ∈ L1 L♯ }, Stab(s, P ∩ M )2 = {(w, γ) ∈ Stab(s, P ∩ M ) | cγ ∈ L1 L♯ }, and to an idempotent eµ2 = X X eµ1 = γ∈X L (s)2 /X L (s)1 eλ⊗γ . γ∈X L (s)2 /(X L (s)∩X G (s,λ) The tower of groups X L (s)1 ⊂ X L (s)2 ⊂ X L (s) corresponds to a tower of K-representations µ1 ⊂ µ2 ⊂ µ, where Vµ2 = L γ∈X L (s)2 /X L (s)1 Vµ1 . Theorem 4.12. The algebra H(G♯ Z(G))s is Morita equivalent with a direct sum of |X L (ω, Vµ )| copies of eµG♯ Z(G) H(G♯ Z(G))eµG♯ Z(G) ∼ = (H(Ts , Ws , qs ) ⊗ EndC (Vµ ))X L (s)/X L (ω,V ) µ (H(Ts , Ws , qs ) ⊗ EndC (Vµ2 ))X ⋊ R♯s ∼ = L (s)2 /X L (ω,V ) µ ⋊ R♯s . HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 51 Here the actions X L (s) and R♯s come from the action of Stab(s, P ∩ M ) described in Lemma 4.10. In particular the group Xnr (L/L♯ Z(G)) acts only via translations on Ts . Proof. By Theorem 3.15 and Corollary 4.9 it suffices to establish the second isomorphism. From Lemma 4.10 we see that Xnr (L/L♯ Z(G)) ∩ Xnr (L, ω) fixes the subalgebra eµ2 H(M )eµ2 ⊂ eµ H(M )eµ pointwise. Hence the same holds for eµ2 ⊗γ H(M )eµ2 ⊗γ = cγ eµ2 H(M )eµ2 c−1 γ for any γ ∈ X L (s). On the other hand, by Lemma 4.11 Xnr (L/L♯ Z(G)) ∩ Xnr (L, ω) does not fix any cγ ∈ L \ L1 L♯ . Therefore M
X L (s)
X L (s)2 L ∼ . (eµ H(M )eµ )X (s) = eµ2 ⊗γ H(M )eµ2 ⊗γ = eµ2 H(M )eµ2 γ∈X L (s)/X L (s)2 The proof of Theorem 4.5.d shows that eµ2 H(M )eµ2 ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ2 ). For (w, γ) ∈ Stab(s, P ∩ M )2 the intertwiner J(γ, ω ⊗ χ−1 γ ) restricts to a bijection → Vµ2 = eµ2 Vw−1 (ω)⊗γ −1 . Vµ2 = eµ2 Vω⊗χ−1 γ Hence the action of Stab(s, P ∩M )2 on H(Ts , Ws , qs )⊗EndC (Vµ ) described in Lemma 4.10 preserves the subalgebra H(Ts , Ws , qs ) ⊗ EndC (Vµ2 ). By equations (158) and (159) the groups Xnr (L/L♯ Z(G)) and X L (ω, Vµ ) act as asserted.  In combination with Lemma 3.5, the above proof makes it clear that the use of the group L/Hλ ∼ = Irr(X L (ω, Vµ )) is unavoidable. Namely, by (16) the operators I(γ, ω ⊗ χ) associated to γ ∈ X L (ω, Vµ ) cause some irreducible representations of G to split upon restriction to G♯ Z(G). But in the proof of Theorem 4.12 we saw that the same operators act trivially on eµ H(M )eµ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ). This has to be compensated somehow, and we do so by adding a direct summand for every irreducible representation of X L (ω, Vµ ). See also Example 5.5. 4.4. Hecke algebras for the derived group. Let ω ∈ Irr(L) be supercuspidal and let σ ♯ be an irreducible subquotient of L ♯ ResL L♯ (ω), or equivalently of ResL♯ Z(G) (ω). Consider the Bernstein torus Tt♯ of t = [L♯ , σ ♯ ]G♯ and Tt of t = [L♯ Z(G), σ ♯ ]G♯ Z(G) . Then Tt♯ is the quotient of Tt with respect to the action of Xnr (L♯ Z(G)/L♯ ). In turn Tt clearly is a quotient of Xnr (L) via χ → σ ♯ ⊗ χ. But it is not so obvious that it is also a quotient of Ts ∼ = Xnr (L)/Xnr (L, ω), ω for χ ∈ X (L, ω) might be complicated. The because the isomorphism ω ⊗ χ ∼ = nr next result shows that this awkward scenario does not occur and that Tt is a quotient of Ts . Lemma 4.13. For ω and σ ♯ as above, σ ♯ ⊗ χ ∼ = σ ♯ for all χ ∈ Xnr (L, ω). 52 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Proof. Let µL be as in (127). By Lemma 4.3 and Theorem 4.5 H(L)sL is Morita equivalent with (164) es H(L)es ∼ = O(Ts ) ⊗ EndC (Vµ ) ⊗ M|L/H | (C). L L λ The analogues of Lemma 3.9 and Proposition 3.8 for L show that H(L♯ Z(G))sL is Morita equivalent with
X L (s) (165) esL H(L)XL (s) esL ∼ ⊗ M|L/Hλ | (C). = O(Ts ) ⊗ EndC (Vµ ) Under the first Morita equivalence, ω is mapped to esL Vω , considered as an O(Ts )module with character ω ∈ Ts . For every γ ∈ X L (ω), ω ⊗ γ is mapped to the same module of (164). Recall the operator I(γ, ω) ∈ HomL (ω ⊗ γ, ω) from (13). It restricts to IesL (γ, ω) ∈ HomesL H(L)esL (esL Vω⊗χ , esL Vω ). Since we are dealing with Morita equivalences and by (15), these operators give rise to algebra isomorphisms (166) Endes H(L)XL (s) es (esL Vω ) ∼ = C[X L (ω, κω )]. = EndL♯ (ω)) ∼ L L It follows from (16) that there exists a unique ρ ∈ Irr(C[X L (ω), κω ]) such that σ♯ ∼ = HomC[X L (ω),κω ] (ρ, ω). Then (166) implies (ρ, es Vω ). es V ♯ ∼ = Hom L L σ But X L (ω ⊗ χ) = X L (ω), κω⊗χ C[X (ω),κω ] L = κω and σ♯ ⊗ χ ∼ = HomC[X L (ω),κω ] (ρ, ω ⊗ χ). Moreover ω and ω ⊗ χ correspond to the same module of (164), so s s ∼ Hom L ∼ s esL (Vσ♯ ⊗χ ) = C[X (ω),κω ] (ρ, eL Vω⊗χ ) = HomC[X L (ω),κω ] (ρ, eL Vω ) = eL Vσ♯ . In view of the second Morita equivalence above, this implies that Vσ♯ ⊗χ ∼ = Vσ♯ as ♯ L -representations.  Now we are finally able to give a concrete description of the Hecke algebras associated to G♯ . Let Ts♯ be the restriction of Ts to L♯ , that is, (167) T ♯ := Ts /Xnr (L/L♯ ) = Ts /Xnr (G) ∼ = Xnr (L♯ )/Xnr (L, ω). s With this torus we build an affine Hecke algebra H(Ts♯ , Ws , qs ) like in (116) and (117). Recall from (26) that there are finitely many Bernstein components t♯ for G♯ such that M ♯ (168) H(G♯ )s = H(G♯ )t . ♯ t ≺s By Lemma 4.13 the Bernstein torus associated to any t♯ ≺ s is a quotient of Ts♯ by a finite group. However, we warn that in general Tt♯ is not equal to Ts♯ , see Example 5.1. We can also describe the 2-cocycle of (169) Stab(ω ⊗ χ, P ∩ M ) := Stab(ω ⊗ χ) ∩ Stab(s, P ∩ M ) coming from Vµ via Lemma 4.10. We note that by (22), (23) and Lemma 2.3.b the group (169) is isomorphic to X G (IPG (ω⊗χ)), via projection on the second coordinate. HECKE ALGEBRAS FOR INNER FORMS Recall the idempotent e♯λ G♯ OF p-ADIC SPECIAL LINEAR GROUPS 53 from (107). We define eµG♯ ∈ H(G♯ ) similarly, as the restriction of eµG : K → C to K ∩ G♯ . Theorem 4.14. The algebra H(G♯ )s is Morita equivalent with e♯λ ♯ H(G♯ )e♯λ G♯ G = M aeµG♯ a−1 H(G♯ )aeµG♯ a−1 ∼ = a∈[L/Hλ ] M|X L (ω,Vµ )| 1 eµG♯ H(G♯ )eµG♯ . There are algebra isomorphisms
X L (s)/X L (ω,Vµ )Xnr (L/L♯ Z(G)) eµG♯ H(G♯ )eµG♯ ∼ ⋊ R♯s = H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ )
X L (s)2 /X L (ω,Vµ )Xnr (L/L♯ Z(G)) ∼ ⋊ R♯s . = H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ2 ) The actions of X L (s) and R♯s come from Stab(s, P ∩ M ) via Lemma 4.10, which involves a projective action on M Vµ = Vµ2 ⊗γ . L L 2 γ∈X (s)/X (s) The restriction of the associated 2-cocycle to Stab(ω ⊗ χ, P ∩ M ) corresponds to the 2-cocycle κI G (ω⊗χ) from (15). Its cohomology class is trivial if G = GLm (D) is split. P Proof. The Morita equivalence, the equality and the first isomorphism follow from Theorem 3.16 and Lemma 4.8. These also show that M|L/Hλ | M|L/Hλ | L eµG♯ H(G♯ )eµG♯ ∼ (eµ H(M )eµ )X (s)Xnr (G) ⋊ R♯s . = 1 1 The proof of Lemma 4.10 can also be applied to the action of Xnr (G) = Xnr (L/L♯ ) on (eµ H(M )eµ ∼ = H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ), and it shows that Xnr (G) acts on via translations of Ts . The remaining action of X L (s)2 is trivial on Xnr (L/L♯ Z(G))X L (ω, Vµ ) by Theorem 4.12. This gives the third isomorphism, from which the fourth follows (again using Theorem 4.12). The definitions (13) and (46) show that the 2-cocycle κ of Stab(ω, P ∩ M ) determined by J(γ, ω ⊗ χ) ◦ J(γ ′ , ω ⊗ χ) = κ((w, γ), (w′ , γ ′ ))J(γγ ′ , ω ⊗ χ) is related to (14) by κ((w, γ), (w′ , γ ′ )) = κI G (ω⊗χ) (γ, γ ′ ). P Let φ be the Langlands parameter of the Langlands quotient of IPG (ω ⊗ χ). Via the local Langlands correspondence κI G (ω⊗χ) is related to a 2-cocycle of the component P group of φ, see [HiSa, Lemma 12.5] and [ABPS2, Theorem 3.1]. Hence κI G (ω⊗χ) is P trivial if GLm (D) is split, and otherwise it reflects the Hasse invariant of D.  Remark. In the case G♯ = SLn (F ) Theorem 4.14 can be compared with [GoRo2, Theorem 11.1]. The algebra that Goldberg and Roche investigate is H(G♯ , τ ), where τ is an irreducible subrepresentation of λG as a representation of P N 1 (KG , λG )∩ G♯ . They show that this is a type for a single Bernstein component t♯ ≺ s. Then H(G♯ , τ ) 54 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD is a subalgebra of our eµG♯ H(G♯ )eµG♯ , because the idempotent eτ is smaller than eµG♯ . In our terminology, [GoRo2] shows that H(G♯ , τ ) ∼ = H(Tt♯ , Ws , qs ) ⋊ Rt♯ , eτ H(G♯ , τ )eτ ∼ = H(Tt♯ , Ws , qs ) ⋊ Rt♯ ⊗ EndC (Vτ ). Only about the part ⋊Rt♯ that Goldberg and Roche are not so sure. In [GoRo2] it is still conceivable that Rt♯ (denoted C there) is only embedded in H(G♯ , τ ) as part of a twisted group algebra C[Rt♯ , δ]. With Theorem 4.14, (actually already with Proposition 3.8) we see that the 2-cocyle δ from [GoRo2, §11] is always trivial. 5. Examples This paper is rather technical, so we think it will be helpful for the reader to see some examples. These will also make clear that in general none of the introduced objects is trivial. Most of the notations used below are defined in Subsections 2.1 and 2.2. Example 5.1 (Torus for s in G♯ smaller than expected). Let σ2 ∈ Irr(GL4 (F )) be as in [Roc, §4]. Its interesting property is X GL4 (F ) (σ2 ) = hχ0 ηi. Here χ0 and η are characters of F × , both of order 4, with χ0 unramified and η (totally) ramified. There exists a similar supercuspidal representation σ3 with X GL4 (F ) (σ3 ) = hχ−1 0 ηi. Let G = GL8 (F ), L = GL4 (F )2 , ω = σ2 ⊗ σ3 . Then X L (ω) = {1, η 2 χ20 } and X L (s) = hηiXnr (L/L♯ Z(G)), Ts = Xnr (L) ∼ = (C× )2 . The natural guess for the torus of an inertial class t♯ ≺ s is Ts♯ = Xnr (L♯ ) = Ts /Xnr (L/L♯ ). Yet it is not correct in this example. Recall from (155) that there exists a χη ∈ Xnr (L) with ω ⊗ η ∼ = ω ⊗ χη . One can check that χη = χ−1 0 ⊗ χ0 6= 1 and X L (s) 6= X L (ω)Xnr (L/L♯ Z(G)). Upon restriction to L♯ , ω decomposes as a sum of two irreducibles, caused by the L-intertwining operator J(η 2 , ω) : ω → ω ⊗ η −2 χ2 ∼ = ω ⊗ η 2 ⊗ (χ−2 ⊗ χ2 ). η σ♯ 0 J(η, ω)2 Let be one of them. We may assume that = up to an unramified twist. Then, with t♯ = [L♯ , σ ♯ ]G♯ : 0 J(η 2 , ω), Tt = Ts /X L (s, σ ♯ ) = Xnr (L♯ Z(G))/hχ−1 0 ⊗ χ0 i, Tt♯ = Xnr (L♯ )/h1 ⊗ χ20 i. Example 5.2 (Ws♯ acts on torus without fixed points). This is the example from [Roc, §4], worked out in our setup. G = GL8 (F ), L = GL2 (F )2 × GL4 (F ) so η stabilizes σ ♯ HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 55 Take η, χ0 , σ2 as in the previous example, and let σ1 ∈ Irr(GL2 (F )) be supercuspidal, such that X GL2 (F ) = {1, η 2 }. Write γ = ηχ0 and ω = σ1 ⊗ γσ1 ⊗ σ2 . Then Ws = 1, R♯s = Ws♯ = W (G, L) =: {1, w}. However, X L (ω) = 1, the action of w on Ts involves translation by γ: (w, γ)·(σ1 χ1 ⊗γσ1 χ2 ⊗σ2 χ3 ) = (γ 2 σ1 χ2 ⊗γσ1 χ1 ⊗γσ2 χ3 ) ∼ = (χ20 σ1 χ2 ⊗γσ1 χ1 ⊗σ2 χ3 ) for χi unramified. In terms of the unramified coordinates, we obtain χγ = (χ20 , 1, 1) = (−1, 1, 1) ∈ (C× )3 , (w, γ) · (χ1 , χ2 , χ3 ) = (−χ2 , χ1 , χ3 ). This is a transformation without fixed points of Ts = Xnr (L), and also of Xnr (L♯ ) and of Xnr (L♯ Z(G)). We have Stab(s) = Stab(ω)Xnr (L/L♯ Z(G)), Stab(ω) = {1, (w, γ)}, and these groups act freely on Ts . It follows that IPG (ω) is irreducible and remains so upon restriction to G♯ . Writing s♯ = [L♯ , ω]G♯ , Theorems 4.12 and 4.14 provide Morita equivalences H(G♯ Z(G))s ∼M O(Ts ) ⋊ Stab(s) ∼M O(Xnr (L♯ Z(G))) ⋊ Stab(ω), ♯ H(G♯ )s ∼M O(Xnr (L♯ )) ⋊ Stab(ω). Example 5.3 (Decomposition into 4 irreducibles upon restriction to G♯ ). This and the next example are
based on [ChLi, §6.3]. Let φ be a Langlands parameter for GL2 (F ) with image { a0 0b | a, b ∈ {±1}} and whose kernel contains a Frobenius element of the Weil group of F . The representation π ∈ Irr(GL2 (F )) associated to φ via the local Langlands correspondence has X GL2 (F ) (π) consisting of four ramified characters of F × of order at most two, say {1, γ, η, ηγ}. The cocycle κπ is trivial, so by (15)   EndSL2 (F ) (π) ∼ = C X GL2 (F ) (π) ∼ = C4 , GL (F ) and ResSL22(F ) (π) consists of 4 inequivalent irreducible representations. Next, let St be the Steinberg representation of GL2 (F ) and consider (170) ω = π ⊗ St ⊗ γSt ⊗ ηSt ⊗ γηSt ∈ Irr(L), where L = GL2 (F )5 , a Levi subgroup of G = GL10 (F ). In this setting Ws = 1, X L (ω) = 1, X G (IPG (ω)) = X GL2 (F ) (π). Identifying W (G, L) with S5 , we quickly deduce Stab(ω) = {1, ((23)(45), γη), ((24)(35), η), ((25)(34), γ)}, and R♯s = Ws♯ ∼ = (Z/2Z)2 . However, the action of Stab(ω) on Ts does not involve translations, it is the same action as that of R♯s . The cocycle κI G (ω) can be deterP mined by looking carefully at the intertwining operators. Only in the first factor of L something interesting happens, in the other four factors the intertwining operators can be regarded as permutations. Hence the isomorphism Stab(ω) → X GL2 (F ) (π) 56 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD induces an equality κI G (ω) = κπ . As we observed above, this cocycle is trivial, so by P (15)   EndG♯ (IPG (ω)) ∼ = C Stab(ω) ∼ = C4 G and ResG G♯ (IP (ω)) decomposes as a direct sum of 4 inequivalent irreducible representations. Theorems 4.12 and 4.14 tell us that there are Morita equivalences H(G♯ Z(G))s ∼M O(Ts )Xnr (L/L ♯ Z(G)) ⋊ R♯s = O(Xnr (L♯ Z(G))) ⋊ R♯s , ♯ H(G♯ )s ∼M O(Ts )Xnr (L/L ) ⋊ R♯s = O(Xnr (L♯ )) ⋊ R♯s . Example 5.4 (non-trivial 2-cocycles). Let D be a central division algebra of dimension 4 over F and recall that D 1 denotes the group of elements of reduced norm 1 in D × = GL1 (D), which is also the maximal compact subgroup and the derived group of D × . Take φ, π, γ, η as in the previous example and let τ ∈ Irr(D × ) be the image of π under the Jacquet–Langlands correspondence. Equivalently, τ has Langlands parameter φ. Then × X D (τ ) = X GL2 (F ) (π) = {1, γ, η, γη}. × As already observed in [Art], the 2-cocycle κτ of X D (τ ) is nontrivial. The group × X D (τ ) has one irreducible projective non-linear representation, of dimension two. Therefore   × EndD1 (τ ) ∼ = C X D (τ ), κτ ∼ = M2 (C) × and ResD1 (τ ) ∼ = τ ♯ ⊕ τ ♯ with τ ♯ irreducible. D Now we consider G = GL5 (D), L = GL1 (D)5 and σ = τ ⊗ 1 ⊗ γ ⊗ η ⊗ γη ∈ Irr(L). This representation is the image of (44) under the Jacquet–Langlands correspondence. It is clear that × X L (σ) = 1, X G (IPG (σ)) = X D (τ ) and Ws = 1, where s = [L, σ]G . Just as in the previous example, we find X L (s) = Xnr (L/L♯ Z(G)) ∼ = Z/10Z, Ws♯ = R♯s ∼ = (Z/2Z)2 ⊂ W (G, L), Stab(σ) = {1, ((23)(45), γη), ((24)(35), η), ((25)(34), γ)}, X G (s) = Stab(σ)X L (s). We refer to Subsection 2.2 for the definitions of these groups. The same reasoning × as for κπ shows that κI G (σ) = κτ via the isomorphism Stab(σ) → X D (τ ). Hence P   G ∼ ∼ EndG (I (σ)) C Stab(σ), κ G (σ) = M2 (C), = ♯ P I G P G ResG G♯ (IP (σ)) and is direct sum of two isomorphic irreducible G♯ -representations. To analyse the Hecke algebras associated to s, we need to exhibit some types. A type for γ as a D × -representation is (D 1 , λγ = γ ◦ Nrd). The same works for other characters of D × . We know that τ admits a type, and we may assume that is of the × form (D 1 , λτ ). It is automatically stable under X D (τ ) and dim(λτ ) > 1 because τ is not a character. Then
(D 1 )5 , λ = λτ ⊗ λ1 ⊗ λγ ⊗ λη ⊗ ληγ HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 57 is a type for [L, σ]L . The underlying vector space Vλ can be identified with Vλτ . We note that M = L and Ts = Xnr (L) ∼ = (C× )5 . Proposition 3.8 and Theorem 4.1 show that there is a Morita equivalence
Xnr (L/L♯ Z(G)) ⋊ R♯s . H(GZ(G))s ∼M O(Ts ) ⊗ EndC (Vλ ) The action of R♯s on EndC (Vλ ) comes from a projective representation of X D (τ ) on Vλτ . It does not lift a to linear representation because κτ is nontrivial. Therefore H(GZ(G))s is not Morita equivalent with × O(Ts )Xnr (L/L ♯ Z(G)) ⋊ R♯s = O(Xnr (L♯ Z(G)) ⋊ R♯s . Similarly the algebras H(G♯ )s and

X (L/L♯ ) (171) O(Ts ) ⊗ EndC (Vλ ) nr ⋊ R♯s = O(Xnr (L♯ )) ⊗ EndC (Vλ ) ⋊ R♯s are Morita equivalent, but (172) ♯ O(Ts )Xnr (L/L ) ⋊ R♯s = O(Xnr (L♯ )) ⋊ R♯s has a different module category. One can show that (171) and (172) are quite far apart, in the sense that they have different periodic cyclic homology. Example 5.5 (Type does not see all G♯ -subrepresentations). Take G = GL2 (F ) and let χ− be the unique unramified character of order 2. There exists a supercuspidal ω ∈ Irr(G) with Xnr (G, ω) = {1, χ− }. Then χ− ∈ X G (ω) and I(ω, χ− ) ∈ HomG (ω, ω ⊗ χ− ). This operator can be normalized so that its square is the identity on Vω . Let G1 be the subgroup of G generated by all compact subgroups. The +1-eigenspace and the −1-eigenspace of I(ω, χ− ) are irreducible G1 -subrepresentations of ResG G1 (ω), and these are conjugate via an element a ∈ G \ G1 Z(G). Any type for [G, ω]G is based on a subgroup of G1 , so it sees only one of the two irreducible G1 -subrepresentations of ω. This phenomenon forces us to introduce the group L/Hλ in Lemma 3.3 (here G/G1 Z(G) ∼ = {1, a}) and carry it with us through a large part of the paper. Example 5.6 (Types conjugate in G but not in G1 G♯ ). Consider a supercuspidal representation ω of GLm (D) which contains a simple type (K, λ). Fix a uniformizer ̟D of D. Assume that there exists γ ∈ X G (s) such that ̟D 1m normalizes K and −1 ·λ∼ ̟D 6 λ. =λ⊗γ ∼ = Then λ and λ ⊗ γ are conjugate in G but not in G1 G♯ . This can be constructed as follows. For simplicity we consider the case where K = GLm (oD ) and λ has level zero. Then λ is inflated from a cuspidal representation σ of the finite group GLm (kD ), where kD denotes the residue field of D. On this group conjugation by ̟D has the same effect as some field automorphism of kD /kF . We assume that it is the Frobenius automorphism x 7→ xq , where q = |kF |. Recall that kD /kF has degree d, so |kD | = q d . We need a σ ∈ Irrcusp (GLm (kD )) such that (173) σ ◦ Frobk ∼ 6 σ, = σ ⊗ γ̄ ∼ = F 58 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD × SLm (kD )) is induced by γ. where γ̄ ∈ Irr(GLm (kD )/kD To find an example, we recall the classification of the characters of GLm (kD ) by Green [Gre]. In his notation, every irreducible cuspidal character of GLm (kD ) is of k [1], where k ∈ Z/(q dm − 1)Z is such that k, kq d , . . . , kq d(m−1) the form (−1)m−1 Im are m different elements of Z/(q dm − 1)Z. × ) with Let kE be a field with q md elements, which contains kD . We identify Irr(kE × × dm dm Z/(q − 1)Z by fixing a generator, a character θ : kE → C of order q − 1. Now ds the condition on k becomes that θ kq 6= θ k for any divisor s of m with 1 ≤ s < m. × Let us call such a character of kE regular. The Galois group Gal(kE /kD ) = hFrobkD i d × ) by θk ◦ FrobkD = θ kq , and the regular characters are precisely those acts on Irr(kE whose orbit contains m = |Gal(kE /kD )| elements. With these notations, [Gre, Theorem 13] sets up a bijection (174) × )reg /Gal(kE /kD ) → Irrcusp (GLm (kD )) Irr(kE θk 7→ σk . × and let fx ∈ GLm (kD ) be such that det(t − fx ) Suppose that x is a generator of kE is the minimal polynomial of x over kD . From [Gre, §3] one can see that Green’s bijection (174) is determined by tr(σk (fx )) = (−1)m−1 θ k (x). (175) Let us describe the effects of tensoring with elements of Irr(G/G♯ Z(G)) and of conjugation with powers of ̟D in these terms. As noted above, the conjugation action of ̟D on GLm (D) is the same as entrywise application of FrobkF ∈ Gal(kD /kF ). With (175) we deduce −1 · σk = σk ◦ FrobkF = σkq . ̟D × )reg /Gal(kE /kD ). This corresponds to the natural action of Gal(kD /kF ) on Irr(kE ♯ Consider a γ ∈ Irr(G/G Z(G)) which is trivial on ker(GLm (oD ) → GLm (kD )). It induces a character of GLm (kD ) of the form γ̄ = γ ′ ◦ NkD /kF ◦ det γ ′ ∈ Irr(kF× ). with We assume that γ ′ = θ|k× . By (175) F

γ̄(fx ) = θ NkD /kF (NkE /kD (x)) = θ NkE /kF (x) = θ x(q dm −1)/(q−1)
= θ (q dm −1)/(q−1) (x). Comparing with (173) we find that we want to arrange that kq ≡ k + q dm − 1 q−1 mod q dm − 1, but kq ∈ / {k, kq d , . . . , kq d(m−1) } mod q dm − 1. For example, we can take q = 3, d = 3 and m = 2. Then q dm = 729, (q dm − 1)/(q − 1) = 364 and suitable k’s are 182 or 182q d ≡ −110. Example 5.7 (Types conjugate in G1 G♯ but not in G1 ). Take G = GL2m (D), L = GLm (D)2 , ω = ω1 ⊗ ω2 with ω1 , K1 , λ1 as in the previous example (but there without subscripts). Also assume that the supercuspidal ω2 ∈ Irr(GLm (D)) contains a simple type (K2 , λ2 ) such that ̟D 1m · λ2 ∼ 6 λ2 . = λ2 ⊗ γ ∼ = HECKE ALGEBRAS FOR INNER FORMS OF p-ADIC SPECIAL LINEAR GROUPS 59 Then (K, λ) = (K1 × K2 , λ1 ⊗ λ2 ) is a type for [L, ω]L . It is conjugate to (K, λ ⊗ γ) −1 1m , ̟D 1m ) ∈ G♯ \ G♯ ∩ G1 , but not by an element of G1 . by (̟D References [Art] J. Arthur, “A note on L-packets”, Pure Appl. Math. Quaterly 2.1 (2006), 199–217. [ABPS1] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “On the local Langlands correspondence for non-tempered representations”, to appear in the Münster Journal of Mathematics, arXiv:1303.0828, 2013. [ABPS2] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “The local Langlands correspondence for inner forms of SLn ”, preprint, arXiv:1305.2638, 2013. [BeDe] J. Bernstein, P. Deligne, “Le ”centre” de Bernstein”, pp. 1–32 in: Représentations des groupes réductifs sur un corps local, Travaux en cours, Hermann, Paris, 1984. [BuKu1] C.J. Bushnell, P.C. Kutzko, “The admissible dual of SL(N ) I”, Ann. Éc. Norm. Sup. (4) 26 (1993), 261–280. [BuKu2] C.J. Bushnell, P.C. Kutzko, “The admissible dual of SL(N ) II”, Proc. London Math. Soc. (3) 68 (1994), 317–379. [BuKu3] C.J. Bushnell, P.C. Kutzko, “Smooth representations of reductive p-adic groups: structure theory via types”, Proc. London Math. Soc. 77.3 (1998), 582–634. [BuKu4] C.J. Bushnell, P.C. Kutzko, “Semisimple types in GLn ”, Compositio Math. 119.1 (1999), 53–97. [Cas] W. Casselman, “Introduction to the theory of admissible representations of p-adic reductive groups”, preprint, 1995. [ChLi] K. F. Chao, W.-W. Li, “Dual R-groups of the inner forms of SL(N )”, preprint, arXiv:1211.3039, 2012. [ChGo] K. Choiy, D. Goldberg, “Transfer of R-groups between p-adic inner forms of SLn ”, preprint, arXiv:1211.5054, 2012. [GeKn] S.S. Gelbart, A.W. Knapp, “L-indistinguishability and R groups for the special linear group”, Adv. in Math. 43 (1982), 101–121. [GoRo1] D. Goldberg, A. Roche, “Types in SLn ”, Proc. London Math. Soc. 85 (2002), 119–138. [GoRo2] D. Goldberg, A. Roche, “Hecke algebras and SLn -types”, Proc. London Math. Soc. 90.1 (2005), 87–131. [Gre] J.A. Green, “The characters of the finite general linear groups”, Trans. Amer. Math. Soc. 80 (1955), 402–447. [HiSa] K. Hiraga, H. Saito, “On L-packets for inner forms of SLn ”, Mem. Amer. Math. Soc. 1013, Vol. 215 (2012). [IwMa] N. Iwahori, H. 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Stevens, “Smooth representations of GL(m, D) VI: semisimple types”, Int. Math. Res. Notices (2011). [Sil] A.J. Silberger, Introduction to harmonic analysis on reductive p-adic groups, Mathematical Notes 23, Princeton University Press, Princeton NJ, 1979. [Sol1] M. Solleveld, Periodic cyclic homology of affine Hecke algebras, PhD Thesis, Amsterdam, 2007, arXiv:0910.1606. 60 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD [Sol2] M. Solleveld, “Parabolically induced representations of graded Hecke algebras”, Algebras and Representation Theory 15.2 (2012), 233–271. [Tad] M. Tadić, “Notes on representations of non-archimedean SL(n), Pacific J. Math. 152.2 (1992), 375–396. [Wal] J.-L. Waldspurger, “La formule de Plancherel pour les groupes p-adiques (d’après HarishChandra)”, J. Inst. Math. Jussieu 2.2 (2003), 235–333. Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., 4 place Jussieu 75005 Paris, France E-mail address: aubert@math.jussieu.fr Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA E-mail address: baum@math.psu.edu School of Mathematics, Southampton University, Southampton SO17 1BJ, England and School of Mathematics, Manchester University, Manchester M13 9PL, England E-mail address: r.j.plymen@soton.ac.uk plymen@manchester.ac.uk Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, the Netherlands E-mail address: m.solleveld@science.ru.nl View publication stats